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A Study of the Motion of Conservative and Nonconservative Dynamical Systems by Means of Field Theory
Chapter 4
A Study of the Motion of Conservative and Nonconservative Dynamical Systems by Means of Field Theory
4.1. Introduction
In this chapter, we consider the problem of integration of the differential equations of motion of conservative and nonconservative dynamical systems by means of certain field theories. In the first part of the chapter, we discuss the Hamilton-Jacobi field method for finding the motion of those dynamical systems that are completely describable by means of a Lagrangian (or Hamiltonian) function. The general idea of this celebrated method is that the problem of integration of the differential equations of motion, which are expressed as a system of ordinary differential equations of the first order (Hamilton’s canonical equations), can be accomplished by finding a complete solution of a single nonlinear partial differential equation of the first order, namely, the Hamilton-Jacobi partial differential equation. More precisely, if we succeed in finding a complete solution of the Hamilton-Jacobi partial differential equation, the motion of the dynamical system can be easily found directly from this complete solution without any additional integration, by using only elementary algebraic operations. Note that the field function figuring in the Hamilton-Jacobi theory is Hamilton’s action, and the differential equations of motion are the characteristics of the HamiltonJacobi partial equation. Naturally, the applicability of the Hamilton-Jacobi method is based upon the premise that it is easier to find a complete solution of the HamiltonJacobi partial differential equation than it is to integrate the ordinary set of the differential equations of motion. Contrary to the widespread opinion that the problem of integrating a partial differential equation is usually more complicated than that of integrating a system of ordinary differential 152
4.2. Hamilton’s Canonical Equations of Motion
153
equations, numerous authors have made it a point to emphasize that the ordinary differential equations of motion “may be difficult to integrate by elementary methods, while the corresponding partial differential equation is manageable” [I]. Recently, Arnol’d [2] made the statement that the Hamilton-Jacobi method is “the most powerful method known for exact integration, and many problems, which were solved by Jacobi, cannot be solved by other methods.” In the second part of this chapter, we make an attempt to establish a field method for studying nonconservative holonomic dynamical systems that do not possess Hamilton’s action integral, and the Hamilton-Jacobi field equation cannot be employed. The basic supposition in this field method is that one of the state variables (a generalized coordinate or a generalized momentum) figuring in the dynamical system can be interpreted as a field depending on time and the rest of the state variables of the dynamical system. The resulting field equation, which we call the basic equation, is a single quasi-linear partial differential equation of the first order. As in the case of the Hamilton-Jacobi theory, the differential equations of motion of the dynamical system are differential equations of the characteristics of the basic field equation. Besides, we will demonstrate that there exists a rather completeformal similarity between the field method presented here and the Hamilton-Jacobi method. Namely, if a complete solution of the basic partial differential equation is available, the motion of the dynamical system can be obtained without any additional integration. Note, finally, that our field method can be applied to conservative dynamical systems (for which the Hamilton-Jacobi method can be also applied) but in this case, the field method is not identical to the Hamilton-Jacobi method. 4.2. Hamilton’s Canonical Equations of Motion Consider a holonomic dynamical system whose Lagrangian function is of the form (4.2.1) L = L(t,x , x), where t denotes time, x stands for the generalized independent coordinates x = (xl, x ” ) , and x = [x’, ..., x”) is the generalized velocity vector.
...,
4.2.1. The Legendre Transformation of L Our first objective is to find a dual (conjugate) function to the Lagrangian function L , which is obtained by means of the Legendre transformation of L with respect to the generalized velocities x i (called the active variables),
154
4. Motion of Conservative and Nonconservative Dynamical Systems
while the generalized coordinates x i and t play a passive role in this transformation. The basic idea of the Lagendre transformation is to find the dual function H(t, x , p ) , named Hamilton's function or the Hamiltonian, in which the partial derivatives of the old (active variables) of the original function L are used as the new independent coordinates p i , namely,
pi =
aL
(i = 1,
..., n).
(4.2.2)
One of the essential points in accomplishing this goal is the requirement to solve (4.2.2) with respect to the old variables .tiand to find n relations of the form
xi = f i(t,x , p ) .
(4.2.3)
Naturally, this inversion is possible under the condition that the following functional determinant (Jacobian) is different from zero, i.e., det()
= det(*)
axi a?
H(t, X , p ) = [pixi - ~ ( Xt ,,XI]
# 0.
i i
=fi(t,x,p)
(4.2.4)
(4.2.6)
(4.2.7) Using (4.2.2) and (4.2.3), the last two terms on the right-hand side of this relation cancel, and we have (4.2.8) These equations are the first group of Hamilton's canonical equations of motion.
4.2. Hamilton's Canonical Equations of Motion
155
It is easy to show, by repeating calculations similar to those used for (4.2.7), that the following relations also follow from (4.2.6): (4.2.9) (4.2.10) It is now straightforward to show that the Legendre transformation is reversible. Namely, if we follow the same procedure as just described, we can demonstrate that for a given Hamiltonian function H = H(t, x , p ) , considering the pi as the active variables, the Legendre's transformation of H = H(t,x, p ) leads to the Lagrangian function for the system, L = L(t, x , x). 4.2.2. The Second Group of Equations
To derive the second group of differential equations of motion, we can proceed in several ways, by using some of the dynamical considerations. For example, we can start from the central Lagrangian equation (which is a modification of D'Alembert's variational principle) discussed in Section 3.10 of Chapter Three. If the dynamical system is subject to nonconservative forces Q;= Qi(t,x , x), then we have, according to (3.10.21), d dt
- @i ax') = 6L
+ Qi 6x' + pi
Note [as we discussed after Equation (3.10.21)] that we are not obliged to suppose that the operators of variation, 6, and differentiation, d / d t , are commutative. Substituting from (4.2.8) into Q; , we express the generalized nonconservative forces in terms of t , x , and p as Q; = Q i ( t , x , p ) . By putting L = pix' - H(t, x , p ) into (4.2.1 l), we obtain
d - ( p i 6 x i ) = 6[piXi - H ( t , x , p ) ] + Q i ( t , x , p ) 6 x i+ pi[(6Xi)*- 6%']. dt (4.2.12) That is to say,
+ p i ( 6 ~ ' ) -p;&' ' + Q;6xi,
(4.2.13)
156
4. Motion of Conservative and Nonconservative Dynamical Systems
or
The group of terms with 6pi vanish because of (4.2.8), and since the 6xi are arbitrary and independent variations, we arrive at the second group of canonical equations: (4.2.15)
The second possibility for deriving these differential equations is to start from Lagrange’s differential equations: (4.2.16)
Using (4.2.2) and (4.2.9) and transforming Qi into Qi, we immediately obtain (4.2.15).
4.2.3. Some Properties of Canonical Equations The canonical, or Hamilton’s canonical, equations of motion, (4.2.17)
form a system of 2n ordinary differential equations of the first order with respect to x i and pi. It is easy to show that if the dynamical system is scleronomic (aH/at = 0) and conservative (Qi = 0), then the Hamiltonian H of the system is a constant of motion. Indeed, differentiating H = H(t,x, p ) with respect to time gives us I? = (aH/axi)xi+ (aH/api)pi, and employing (4.2.17) (with Qi = 0), we have I? = 0, or H(t,x , p ) = const.
(4.2.18)
Note that, according to Equations (4.2.5) and ( 3 . 7 4 , this conservation law is identical with the Jacobi integral discussed in Section 3.7-A of the previous chapter. Also, if Hamilton’s function and the generalized forces Qi do not depend on a particular generalized coordinate, say x s (s is a fixed integer), it follows from the second group of canonical equations in (4.2.17) that p s = const.,
which is the cyclic conservation law discussed in Section 3.7.2.
(4.2.19)
4.2. Hamilton's Canonical Equations of Motion
157
Now we discuss the question of derivation of the canonical equations of
motion,
(4.2.20)
from Hamilton's variational principle. It is well known that, if we wish to derive the Lagrangian equations of motion, we can use Hamilton's action in the form
Z=
1
t1
L(t,x,x)dt,
(4.2.21)
to
and Hamilton's principle asserts that Z is stationary for the actual path of motion. This implies that the first variation of the action integral along the real path vanishes, 61 = 0, which leads to Lagrange's equation, (4.2.16), with Qi = 0. Using relation (4.2.5), we can express L ( t , x , x) in terms of the canonical variables, and Hamilton's action reads
Z=
S
tl
to
[Pix' - H(t, x , P ) ] dt.
(4.2.22)
The variation of this integral demands some care. In Hamiltonian dynamics, the generalized coordinates x i and the generalized momenta pi play an equal role. If we wish to vary the action, (4.2.22), we have to take into account that the variables x i and pi are not independent, but that they have to satisfy the (nonholonomic) constraints (4.2.23)
which arise from the definitions of H a n d p , Equations (4.2.5) and (4.2.2). As proposed by J. R. Ray (see [3]), we can vary x i and pi independently in (4.2.2), after introducing the constraints (4.2.23), by means of Lagrange multipliers Li (i = 1, ..., n). The modified action integral then becomes
=
1
fl
(4.2.24)
Ll(t, x , x, p , p , A ) dt.
to
It is a standard supposition to assume that the generalized coordinates are specified over a given interval of time, that is, that xi(to)= ai,
xi(tl) =
bi
( a ,bi constants).
158
4. Motion of Conservative and Nonconservative Dynamical Systems
[This supposition was tacitly assumed in the variational formulation of Lagrange's problem, (4.2.21).] Then varying (4.2.24), we easily conclude that the variational equation 61* = 0 is equivalent to the Euler-Lagrange equations :
d aL1 aL1 - 0, dt api api
(4.2.25) (4.2.26)
aL aAi
-=
(4.2.27)
0.
Hence we find, respectively, (4.2.28)
(pi + Ai)'
aH +7 + Ajax
a2H ax' apj
=
0,
(4.2.29) (4.2.30)
Combining (4.2.28) and (4.2.30), we have (4.2.31) For nondegenerative dynamical systems, we must have det(*) api apj # 0,
(4.2.32)
and therefore, from (4.2.31), we find that all Lagrangian multipliers are zero : Ai = 0. (4.2.33) The requirement (4.2.32) is necessary in order that the first group of the canonical equations, .ti= aH/api implicitly define the pi as functions of xi, xi,and t. Thus, li = 0 indicates that even if we ignore the constraint (4.2.23) and vary the action integral (4.2.22), considering x i and pi as independent variables, we will arrive at the canonical differential equations of motion. This shows that the usual assumption of ignoring the contraints implied by
4.2. Hamilton's Canonical Equations of Motion
159
equation (4.2.23) is indeed justified, since the variational equations (4.2.28) and (4.2.29) produce the correct differential equations for Li = 0. Next let us note that the canonical differential equations of motion (and also Lagrangian equations), can be derived from the standpoint of optimal control theory (see [4]). To show this, let us start from Hamilton's action, (4.2.21), and let the generalized velocities X i be nominated as the control variables during the motion of the system, so that 2i
(4.2.34)
= ui
Now, in the terminology of optimal control theory, the action integral (4.2.21) is the cost function, while Equations (4.2.34) can be interpreted as the differential equality constraints. Extremizing the cost function (4.2.21) subject to the constraints (4.2.34) yields the same result as that obtained by extremizing the modified action (see [ 5 ] ) :
I
1
fl
=
[-L(t,X,
to
U)
+ pi(Ui - Xi)] dt,
(4.2.35)
where the pi stand for the Lagrange multipliers. As usual in optimal control theory, we define Hamilton's function to be H(x, U , P , t)
L(t, X , u).
(4.2.36)
[H(t,X , p , U ) - pixi] dt.
(4.2.37)
= pixi -
Thus, the -modified action becomes
I
=
it' to
The variation of I,considering xi,pi and ui as independent variables, yields
(4.2.38)
Assuming that axi = 0 at t = to and t = tl , the condition 61= 0 is equivalent to (4.2.39)
These equations are the equations of motion for a mechanical system and are equivalent to Lagrange's equations and Hamilton's canonical equations. In fact, from (4.2.36), we find that
160
4. Motion of Conservative and Nonconservative Dynamical Systems
and, combining these results with (4.2.39), we obtain (4.2.41)
as well as Lagrange's differential equations of motion, (aL/ax')' aL/ax' = 0. From (4.2.41), we observe that the Lagrange multiplierspi are, in fact, the generalized momenta. The equations of motion (4.2.39) consist of 2n first-order differential equations and n algebraic equations [the third system in (4.2.39)]. In optimal control theory, this algebraic system of equations is called the extremizing condition for the Hamiltonian. Here, this condition yields the relations p; = aL/ax' between the generalized momenta and the generalized velocities. However, such relations were introduced in Hamiltonian mechanics by Legendre's transformation. Finally, let us consider, as a simple illustration, the motion of a heavy particle of mass m constrained to move on the inner surface of a vertical circular cylinder of radius R under the action of a constant gravitational field. At t = 0, the particle is projected in such a way that the initial velocity vector VO makes an angle a with the y-axis (see Figure 4.2.1). Taking the polar angle 8 and vertical distance z for the generalized coordinates, we have x = R cos 8, y = R sin 8, z = z , and kinetic energy of the particle is m m T = - (x2 + 3' + i 2 )= - (R'P + i2). 2 2 Since the potential energy of the particle is
ll = -rngz,
Fig. 4.2.1. Motion of a heavy particle inside a vertical cylinder.
161
4.2. Hamilton’s Canonical Equations of Motion
where g is the acceleration of gravity, the Lagrangian function takes the form L = T - ll
=
m -(R2b2 2
+ i Z+) mgz.
(4.2.42)
The generalized momenta are defined by aL P O= ae = mRZb,
pz = mi.
(4.2.43)
Solving with respect to the generalized velocities, we have the first group of canonical equations : .
1
e = - mR2~e ,
1
i = -Pz.
m
(4.2.44)
The Hamiltonian of the system is defined, according to (4.2.5), to be
H
= p0b
m
+pzi - (R2e2 + i Z) mgz. 2
Substituting (4.2.44) into this relation, we have H=-
1
(4.2.45)
and 6 is a cyclic coordinate. Therefore, the second group of canonical equations is p 0
p z = mg.
= 0,
Hence, p0 = CI,
p z = mgt
+ CZ.
(4.2.46)
Substituting this into (4.2.44), and using the fact that Rb(0) = uo cos a and
i ( 0 ) = v ~ s i n a we , find that
CI = mRvo cos a,
C2 = mvo sin a.
(4.2.47)
Integrating (4.2.44), we finally find that the motion of the particle is given by 8 = (vo/R)tcosa, z = (gt2/2) + votsina, (4.2.48) p0 = mRvo cos a, p z = mgt + mvosina.
162
4. Motion of Conservative and Nonconservative Dynamical Systems
4.3. Integration of Hamilton's Canonical Equations by Means of the Hamilton-Jacobi Method In this section, we focus on Hamilton's canonical differential equations of motion for the case of potential holonomic dynamical systems,
and demonstrate that the problem of integrating these equations can be replaced by an equivalent problem of finding a complete solution of a nonlinear partial differential equation of the first order, called the HamiltonJacobi equation. To show this, we first introduce a scalar field function.
s = S(t, X I , ...,X " ) ,
(4.3.2)
which depends on the time t and the generalized coordinates x i . Let us define the generalized momentum vector of the dynamical system pi to be the gradient of this scalar field function:
as
(4.3.3)
P i = g -
Our aim is to obtain the differential equation that satisfies the field function S(t, x), where x = [ X I , ...,x " ) . Naturally, this differential equation should emanate from the dynamical characteristics of the material system in question. In order to involve the dynamics, we start from the central Lagrange equation, (4.2.1I), considered in Section 4.2, in which we suppose that (a) Qi = 0, and (b) the symbols of variation and differentiation are commutative:
6d - d6
=
0,
(4.3.4)
and that the time is not varied: 6t = 0. Thus, we consider the variations as contemporaneous (simultaneous) Lagrangian variations. Therefore, we have (4.3.5) where L
=
L(t,x,%)is a Lagrangian function for the dynamical system.
163
4.3. Integration of Hamilton's Canonical Equations
Passing to the canonical variables, i.e., using the relation (4.2.5), we have
d
(4.3.6)
(pi 6~')= Li[piXi- H(t, X , P I ] .
Employing the supposition (4.3.3), we find that
;(-$ax')
=
s[ - $ X i
-
..., x",,l,as ...,ax" E)]
H(t,.l,
(4.3.7)
Adding and subtracting the term aWat on the right-hand side, we obtain
From this equation it follows that if we take (4.3.9)
the rest of (4.3.8) becomes the identity
d dt
-(as)
=
ss
which is, of course, compatible with (4.3.4). The partial differential equation (4.3.9) is the Hamilton-Jacobi equation with respect to the field function S(t, x l , ...,x"). In order to interpret the field function in more familiar way, we again use (4.2.5), L(t, X , X) = pixi - H(t, X , p ) . (4.3.10) Adding and subtracting the term as/& on the right-hand side of this equation and using (4.3.3) and (4.3.9), we find that
d W ,x) L(t,x,X) = ___ dt
(4.3.11)
dS = L dt
(4.3.12)
or Therefore, the field function S = j L dt can be interpreted as Hamilton's action integral. Let us suppose that we know a complete solution of the HamiltonJacobi equation (4.3.9), a solution that is an expression of the form
s = S(t,x', ..., X " , c1, ..., C,) + C"+l.
(4.3.13)
164
4. Motion of Conservative and Nonconservative Dynamics1 Systems
where the C's are arbitrary constant parameters. A complete solution of the Hamilton-Jacobi equation is understood to be an expression of the form (4.3.13) that contains the same number of constant parameters Ci as the number of independent variables, t , xl, ...,x", and that, when substituted into (4.3.9), reduces it to an identity. There is another important property of every complete solution of the Hamilton-Jacobi partial differential equation: if we eliminate the constants CI, . . ., Cn+l from (4.3.13) and from the n + 1 equations
pi =
as ax
= f i ( t , X 1..., , x",
~
1
..., , Cn)
(i = 1 ,
..., n), (4.3.14)
as - = F(t, X I , ...,x", CI, ..., Cn), at
we should obtain the Hamilton-Jacobi equation, (4.3.9). However, since the function S does not enter directly into the HamiltonJacobi equation, and since only the partial derivatives of S figure inside it, we can see that one constant should appear additively in the complete solution, as indicated in (4.3.13). Now it is clear also that, for the elimination of C1, ..., C, , and for the purpose of obtaining the Hamilton-Jacobi equation as the result of this elimination, we need only the system (4.3.14). Thus, the expression (4.3.13) is redundant for this purpose. In fact, we will demonstrate that the additional constant C n + l is not an essential parameter in the Hamilton-Jacobi method. Note also that the term aS/at enters linearly in the Hamilton-Jacobi equation. Therefore, in order to obtain the Hamilton-Jacobi equation by elimination of CI ..., C,, in the system (4.3.14), we need only to calculate these parameters from this first group of equations in (4.3.14) and then to substitute them into the second relation in (4.3.14). However in order to be able to calculate CI ..., Cn, we have the condition that the Jacobian determinant be different from zero in the first group of equations in (4.3.14): (4.3.15)
We will now discuss the famous Jacobi theorem, which states that if we have a complete solution of the Hamilton-Jacobi equation of the form (4.3.13), then the solution of the canonical system of the differential equations of motion, (4.3.1), is determined by the following relations:
as
P i = i , ax
as
-=
aCi
Bi
(i = 1,
..., n),
(4.3.16)
4.3. Integration of Hamilton’s Canonical Equations
165
where the Bi (i = 1, ..., n) are a new set of arbitrary constant parameters. The first group of equations in (4.3.16) determine functions of the form pi = Pi(t,x 1 , ...,x “ , C1, ..., Cn).
(4.3.17)
Thus they determine the generalized momenta of the dynamical system in terms of the time t and the generalized coordinates x i . The second group of equations in (4.3.16) give equations of the form
Bi = Bi(t,X1,..., x,, C1, ..., C,)
( i = 1, ..., n).
(4.3.18)
By solving this algebraic system with respect to the x i , we obtain a solution of the dynamical system in the form
where we find the generalized coordinates as functions of time t and the 2n arbitrary constants Ci and Bi. We will also suppose that Equations (4.3.17) are solvable with respect to the parameters Ci, where
Wi(t,X , ..., x n , p l ,...,pn) = Ci 1
(i = 1, ..., n ) .
(4.3.20)
These equations are the conservation laws of Hamilton’s canonical equations (4.3.1). The proof of the Jacobi theorem consists of demonstrating that the solution (4.3.16) satisfies the canonical differential equations of motion (4.3.1) identically. To show this, we start with the Hamilton-Jacobi partial equation (4.3.9), into which we substitute a complete solution to obtain an identity of the form (4.3.21)
The differentiation of (4.3.21) with respect to the various parameters will produce the new identities. First differentiating (4.3.21) partially with respect to the Ci’S, we have
a2s + -aH a2s(t,X , c) = 0
ataci
apj
ax-l aCi
( i , j = 1, ..., n).
(4.3.22)
Taking the total time derivative of the second group of equations in
166
4. Motion of Conservative and Nonconservative Dynamical Systems
(4.3.16), we find that
=0
( i , j = 1, ..., n).
(4.3.23)
These last two systems of equations represent a nonhomogeneous system of linear equations with respect to aH/apj and x’, respectively. The coefficients of these two systems are equal, and, according to (4.3.19, are different from zero. Therefore the roots of both systems have to be identically equal, that is, ..
x’
aH api
-
( i = 1, ..., n).
(4.3.24)
This means that the first group of Hamilton’s canonical differential equations, (4.3. l), are identically satisfied. . To prove that the second group of canonical equations (4.3.1) are identically satisfied, we find the partial derivatives of the identity (4.3.21) with respect to the Xi’s:
a2s(t,x,c) aH a2s(t,x,c) . + i + kJ = 0. at axi ax axi axj
(4.3.25)
We compare this equation with the total time derivative of the first group of equations in (4.3.16): pi = a2s(t,X , c)& azs(t,X , c) axi ad axi at +
(4.3.26)
Substituting (4.3.26) into the identity (4.3.25), we prove that the second group of canonical equations,
aH p i 1 -7, ax
(4.3.27)
is also identically satisfied, which completes the proof of the Jacobi theorem. Therefore, we can conclude that the problem of integration of the canonical equations of motion (4.3.1) can be completely replaced by the problem of finding a complete solution of the Hamilton-Jacobi partial differential equation, (4.3.9). This means that if a complete solution of this equation is available, we can find the motion of the dynamical system without any additional integration by using only the operations of simple partial differentiation and algebra.
4.3. Integration of Hamilton’s Canonical Equations
167
It is of importance to note that there exists an intimate connection between the canonical system (4.3.1) and the Hamilton-Jacobi partial differential equation, a connection that can be traced in the opposite way (direction) from the Jacobi theorem previously stated. In fact, it was also stated by Jacobi that when we know a general solution of the canonical system . aH aH (4.3.29) XI=-, , . - - -=ip api we can always find a complete solution of the Hamilton-Jacobi partial differential equation
at
(4.3.30)
(see, for example, [l]). In other words, finding a general solution the system (4.3.29) and finding a complete solution of the Hamilton-Jacobi partial differential equation (4.3.30) are mathematically equivalent problems. Using the terminology of the theory of first-order partial differential equations, we can say that the ordinary differential equations (4.3.29) are the differential equations of the characteristics of the partial differential equation (4.3.30). It is of interest to point out that in recent years, in many branches of mechanics (especially cellestial mechanics), engineering and optimal control theory, the Hamilton-Jacobi method and the other related field methods have been intensively developed (see, for example, [6] and [7]) and have been applied to many practical problems (see the discussion in the introduction of this Chapter). 4.3.1. Examples
THEHARMONIC OSCILLATOR In order to illustrate the foregoing theory, we consider the elementary problem of a linear harmonic oscillator, namely, a material point of mass m that moves along a straight line (the coordinate x direction) in an elastic field whose elastic coefficient is c. In this case, it is easy to show that Hamilton’s function is 1 1 H = - p 2 + -mo2x2, (4.3.31) 2m 2 where o
=
( ~ / r n ) ’ ’is~ the circular frequency.
168
4. Motion of Conservative and Nonconservative Dynamical Systems
The Hamilton-Jacobi partial differential equation is, therefore, (4.3.32)
To find a complete solution of this equation, consider a function S of the form
s = $X2f(t),
(4.3.33)
where f ( t ) is an unknow function of time. Then the problem is reduced to a Riccati equation: f + (l/m) f + mw2 = 0, whose general solution is f = - mw tan(wt + C ) , where C is a constant. Therefore, a complete solution of (4.3.32) is
S = -tmwx2 tan(wt
+ C ) + D,
(4.3.34)
where D is an arbitrary additive constant that plays no essential role in further considerations. According to the Jacobi theorem, we find that
as
+ C)
p = - = -mwxtan(wt
ax
and
as -_ _
ac -
mwx2 2cos2(wt +
c)= B .
(4.3.35)
(4.3.36)
From the last equation, we find the motion:
x=
(-g) 1/2
cos(wt
+ C).
(4.3.37)
Substituting this into (4.3.39, we can find the generalized momentum: p = - (- 2 m ~ B ) ”sin(wt ~
+ C),
(4.3.38)
which completes the motion of the system. Note also that (4.3.35) implies a conservation law of the form wt
+ arctan(p/mwx)
= const.
(4.3.39)
THE LINEARLY DAMPED OSCILLATOR The linearly damped, one-dimensional harmonic oscillator is governed by the differential equation of motion
R
+ 2kX + wo2x = 0,
(4.3.40)
4.3. Integration of Hamilton's Canonical Equations
169
whose Lagrangian function is L = +(? - o O 2 x 2 p .
(4.3.41)
The canonical momentum is
p =x2kf,
(4.3.42)
and Hamilton's function becomes
H = +(pZe-zk' + &X2e2k').
(4.3.43)
The Hamilton-Jacobi equation thus takes the form (4.3.44)
For a complete solution of this equation, let us seek a function S of the form
~ ( xt ), = +x2f(t)e2k',
(4.3.45)
where f ( t )is to be determined. Substituting this last relation into (4.3.44), we find the ordinary differential equation for f ( t ) :
+f 2 +
f + 2kf
= 0,
(4.3.46)
+ C),
(4.3.47)
which has the general solution
f = -k
-
otan(ot
where C is a constant of integration and
00"
-
k2 =
(4.3.48)
0'.
Thus, the complete solution of (4.3.44) is found to be
S = - i x 2 [ k + o tan(ot
+ C)]2k'.
(4.3.49)
Applying the Jacobi theorem, we have
as
+ o tan(ot + C)]2k'
(4.3SO)
meZk' = B = const., 2 cos2(ot + C )
(4.3.51)
p = - = -x[k ax and
as -- - _ x2 _ ac
which means that
x
=
( - 2 B / o ) 1 ' 2 e - k r ~ ~+~ (Co)t.
(4.3.52)
This is the well-known solution of the linearly underdamped oscillator, (4.3.40). Inserting (4.3.52) into (4.3.50), gives us the generalized momentum
170
4. Motion of Conservative and Nonconservative Dynamical Systems
as a function of time: p
=
k(- 2 ~ / w ) ' / ~cos(wt e~'
+ C ) + (- 2 ~ o ) " ~ sin(ot e ~ ' + C)
(4.3.53)
It is interesting to note that the same problem was treated by Denman and Buch [8] by supposing that the complete solution of the Hamilton-Jacobi equation, (4.3.44), is of the form S ( t , x ) = - Q ( x ) ~ ~In' . this case, the problem is reduced to the differential equation
@y
-
2kQ
+ mix2 = 0,
which cannot be integrated in a closed form. The authors applied an implicit procedure in the application of the Jacobi theorem, which makes the process of finding the motion rather complicated. THEFORCEDOSCILLATOR Let us also consider the forced oscillatory motion of a dynamical system with one degree of freedom, whose differential equation of motion is X
+ O ~ X= h cos a t ,
(4.3.54)
where h, w , and S2 are given constant parameters and w # S2. It is easy to verify that this equation can be interpreted as the Lagrangian equation
for the Lagrangian function of the form L = - x + hS2 sin:;y 2 w2-
-2;
(. -
h cos S2t w2 -
n2
.
(4.3.55)
Introducing the momentum, p = -aL= x
(4.3.56)
ax
the Hamiltonian function becomes +-0
2
2(
x - hCosQr)i. w 2 - a2
(4.3.57)
17 1
4.3. Integration of Hamilton’s Canonical Equations
The canonical equations of motion are then
(4.3.58)
To find a general solution of this system, we turn to the Hamilton-Jacobi equation : o2-
n2
We try to find a complete solution of this partial differential equation of the form (4.3.60)
where f(t) is an unknown function. Substituting this last expression into (4.3.59), we obtain o2= 0 and thus
f=
- 0 tan(ot
+ C).
f +f + (4.3.61)
Therefore the complete solution of (4.3.59) is (4.3.62)
Applying the Jacobi theorem, we find that h cos nt o2-
and
as ac
2
o2-
n2
n2 tan(cot + C ) 1
cos2(ot + C )
= B = const.
(4.3.63)
(4.3.64)
From this last equation, it follows that the motion of the system is cos(0t
+ C ) + o2-h n2cos n t .
(4.3.65)
Taking into account this expression, the momentum (4.3.63) becomes p
= - (-
2 ~ 0 ) sin(cot ” ~
+ C)
and the solution of the canonical system (4.3.58) is complete.
(4.3.66)
172
4. Motion of Conservative and Nonconservative Dynamical Systems
It is of interest to give a brief comment concerning the choice of the Lagrangian function (4.3.55). In fact, as shown in Reference [9], the expression 1 2
- (x
; .(
+ hRsin;;Y o2-
+-0
x-
‘0s
o2-
“7 R2
= D = const.,
(4.3.67)
which differs slightly from the Lagrangian (4.3.59, is a quadratic conservation law of the differential equation, (4.3.54). Thus, the Lagrangian function (4.3.55) was selected by the analogy that exists between the quadratic first integrals of energy type and the Lagrangian functions. Strictly speaking, this analogy is valid only for purely conservative cases but, fortunately, it can be used here by chance. Similarly, for the case of resonance, the differential equation of motion X
+ O’X
=
h cos at
(4.3.68)
admits a quadratic conservation law of the form
’(. Fcoscot 2
-
-
(4.3.69)
It is easy to verify that Lagrange’s equation will produce (4.3.68) for the Lagrangian function
which will be taken as the basis of further consideration. The Hamiltonian function is found to be H = -1p 2 2
+ (ihtcosut + (4.3.71)
and the canonical equations are x=p
p =
h ht + -coswt + -sinot, 2 20
-02’
(
;:
x - -shot
).
(4.3.72)
4.3. Integration of Hamilton’s Canonical Equations
173
The Hamilton-Jacobi partial equation thus becomes
(4.3.73)
whose complete solution is of the form
( Y
y
(
)
S = -to x
-
- sin cot tan(wt + C ) ,
(4.3.74)
where C is a constant. Applying the Jacobi theorem, we find that p = - w x - - ht sinwt tan(wt 20 and
1
+ c)
= B = const.,
(4.3.75)
(4.3.76)
which means that cos(ot
ht + C ) + -sin 20
at.
(4.3.77)
This is the familiar solution for the resonant case. THEFORCED, LINEARLY DAMPEDOSCILLATOR Finally, let us consider the forced, linearly damped oscillator with a single degree of freedom, whose differential equation of motion is X
+ 2kX + O ~ X= h c o s a t .
(4.3.78)
At the same time, consider the Lagrangian function
L = (i[X
+ SZ(A sin SZt - B cos at)]’ - * w i [ x - ( A cos a t + B sin at)]’)2k‘,
(4.3.79)
where A and B are constants to be determined in such a way that the Lagrange equation (d/dt)(aL/aX) - (aL/ax) = 0 generates the correct differential equation of motion, (4.3.78). Inserting (4.3.79) into this Lagrange
174
4. Motion of Conservative and Nonconservative Dynamical Systems
equation, we obtain (x + 2kx
+ o0”x + cosSZt[-A(o0” - a’) - 2kSZBI + sin at[-B(o0” - a’) + 2 k M ] ) 2 k ‘= 0.
(4.3.80)
Therefore, if A and B are selected such that
a’) + 2kSZB = h, 2 k M - B(& - a’) = 0,
A(O: -
(4.3.81)
or A =
(00”
a’) a’)’ + 4k’SZ’ ’
h(& -
B =
(00”
2kha
-
(4.3.82)
a’)’ + 4k’SZ’ ’
then Equation (4.3.80) does, indeed, reduce to (4.3.78). We will now take the Lagrangian function (4.3.79) as the basis for constructing the Hamilton-Jacobi partial differential equation. From Equation (4.3.79), we find Hamilton’s function in the form H = *p2e-2kt- pQ(A sin SZt - B sin at)
+ *o0”[x - (A cos SZt + B sin SZt)]22k‘.
(4.3.83)
The canonical differential equations are then 3 = pe-2k‘ - SZ(A sin SZt - B cos at),
p = - o i [ x - (A cos SZt
+ B sin SZt)]2k‘,
(4.3.84)
where A and B are given by (4.3.82). The Hamilton-Jacobi equation is
+ -21 o z [ x - (A cos SZZt + B sin SZt)]z2k‘ = 0.
(4.3.85)
Let us try to find a complete solution of this equation of the form
S = *[x
-
(A cos SZt
+ B sin SZt)]’eZk‘f(t),
(4.3.86)
where f(t) is an unknown function. Substituting Equation (4.3.86) into (4.3.85), this expression is reduced to f + 2kf + f + 00” = 0. Integrating, we find that
f=
- [k
+ o tan(ot + C ) ]
(C = const.),
(4.3.87)
175
4.4. Separation of Variables in the Hamilton-Jacobi Equation
with (4.3.88)
Therefore,
s = -1 [x - ( A cos SZt + B sin SZt)I2[k+ o tan(ot + C)]e2k'. (4.3.89) Applying the Jacobi theorem, we find the generalized momentum p to be
as
p = - = - [x - ( A cos SZt ax
+ B sin SZt)][k+ o tan(ot + C)]e2k', (4.3.90)
and
- $[x - (A cos Qt
+ B sin Qt)I2cos2(ot + C ) = B = const.
(4.3.91)
From Equation (4.3.91), it follows that x = (- 2 B / ~ ) ' / ~ ecos(ot -~'
+ C ) + A cos SZt + B sin SZt (4.3.92)
which is the well-known solution of (4.3.78). Inserting this result into (4.3.91), we find the momentum as a function of time: p = - ( - 2 B / o ) ' / 2 e k t [ osin(wt
+ C ) + kcos(ot + C ) ] ,
(4.3.93)
where B and C are arbitrary constants.
4.4. Separation of Variables in the Hamilton-Jacobi Equation
Although it is not possible to find any universal method for constructing a complete solution of the Hamilton-Jacobi partial differential equation, there are a number of cases in which finding a complete solution is straightforward; in such cases, obtaining the motion of a dynamical system is simpler by the Hamilton-Jacobi method that integrating the differential equations of motion directly. Very frequently, under certain conditions, it is possible to separate the variables of the Hamilton-Jacobi equation and the problem is reduced to quadrature. For example, one of the commonIy occurring cases where finding a complete solution is considerably simplified is the case of a purely
176
4. Motion of Conservative and Nonconservative Dynamical Systems
conservative dynamical system, for which Hamilton's function does not depend explicitly on time t. The Hamilton-Jacobi equation then becomes
s, ...,
(4.4.1)
+ Si(X', ...,x"),
(4.4.2)
..., x",
at
By introducing the transformation
S = -Et
where E is a constant, Equation (4.4.1) reduces to the form (4.4.3)
It is clear that the constant E has the interpretation of total mechanical energy of the dynamical system [see (4.2.18)]. Suppose we find a complete solution S of the partial differential equation (4.4.3), which, in addition to x ' , ...,x" contains the constant CI = E and (n - 1) constants CZ,..., Cn :
5'1
= Sl(x',
Then (4.4.2) becomes S = -Et
..., x", E, C2, ..., Cn).
(4.4.4)
+ S l ( X ' , ...,x",E, C2, ..., Cn).
(4.4.5)
Applying the Jacobi theorem, (4.3.16), we have that the motion of the dynamical system is determined by the relations
as
-=
aE
- t + -asl =B aE
1
- -to,
as, ac,
-=
B,,
a = 2,3,
..., n.
(4.4.7)
Next, if some of the generalized coordinates are cyclic (ignorable), then the method of separation of variables can be advantageously applied. Let us consider the case for which n - k independent coordinates are ignorable, in that they do not appear explicitly in H . The Hamilton-Jacobi equation is of the form
as
,...,x k , s ,...,
(4.4.8)
The complete solution of this equation can be expressed as a linear function relative to all of the ignorable coordinates, namely, S = s ~ ( x ' , ..,x k , ~
1
..,., Cn, t ) + C ~ X ,
(a = /t
+ 1, ...,n).
(4.4.9)
4.4. Separation of Variables in the Hamilton-Jacobi Equation
177
From this equation we have
as -- -as2 _ at
as
at
-=
ax,
’
C,
as
as,
ax1
axi
.-
(a = k
(i = 1,2, ...)k);
+ 1, . . . , T I ) .
(4.4.10) (4.4.11)
From (4.4.11), on the basis of (4.3.16), it follows that n - k momenta are constants. Therefore, the Hamilton-Jacobi equation (4.4.8) becomes
If a complete solution of this partial differential equation is available, then by applying the Jacobi theorem, we can easily find the solution of the corresponding dynamical system. Finally, let us consider separation of variables for dynamical systems whose Hamiltonian functions are of the form H = H [ t ,X I ,
...,x k ,PI, ...,Pk,fk+l(Xk+*,pk+l),-. ,fn(x”,Pn)],
(4.4.13)
depends only on the corresponding single pair where every function (couple) of canonical variables x i , pi. The Hamilton-Jacobi equation corresponding to (4.4.13) is
=o
(4.4.14)
For a complete solution of this equation, we seek a function S of the form S = V(X’,...,x k , C1, ..., Cn, t )
n
+ C
j=k+l
W(X’, C1,
- 9
Cn), (4.4.15)
where wj is the additive part of the action depending only on the generalized coordinates XJ and all constant parameters Ci(i = 1 , ...,n). From (4.4.15), we have
as -- av _ at at ’ -as -
ax,
- -aw, ax,
as
av
a x i - axi (a = k
(i = 1 ,
+ 1, ..., n),
...,k),
(4.4.16) (4.4.17)
178
4. Motion of Conservative and Nonconservative Dynamical Systems
and the partial differential equation (4.4.14) becomes
av
t , X 1 , ..., x k - - . - 7axk "",fk+l(x
at =
k+l
Y
awk+l
**-,f"(x"7$)]
(4.4.18)
0.
If the expression (4.4.15) is indeed a complete solution of the HamiltonJacobi equation (4.4.14), then the partial differential equation (4.4.18) should become an identity for arbitrary values of the generalized coordinates x k + ' , ...,x". However, this is possible only under the condition that all functions fi ( x i , a W / a x ' ) ( i = k + 1, ..., n) have a constant value: (a = k
+ 1, ..., n).
(4.4.19)
This is a system of n - k ordinary differential equations. Now, the Hamilton-Jacobi equation, (4.4.18), becomes -
at
+H
(
XI,
av ...,X k Y
av aXk'
If we are able to find a complete integral of this partial differential equation and to integrate (4.4.19), then, using (4.4.16), (4.4.17), and the Jacobi theorem, we can find the solution of (4.4.14) in the form Pi =
av a wa
Pa = axa
as _ - Bi aCi
( i = 1,
...)k),
(a = k
+ 1, ..., n),
(4.4.21)
(i = 1, ..., n).
The method of separation of variables (coordinates) we have just considered includes the ignorable coordinates as a special case. In fact, by putting fO1(xa,pa) = p a (a = k + 1, ...,n) in (4.4.13), the system (4.4.19) becomes
a wa - Ca
-ax.
(a = k
+ 1, ..., n),
(4.4.22)
and the complete integral (4.4.15) is identical with the complete solution (4.4.9) considered before. The separation of variables in the Hamilton-Jacobi theory can be considered, in some sense, to be the generalization of the notion of cyclic
4.4. Separation of Variables in the Hamilton-Jacobi Equation
179
(ignorable) coordinates and their conservation laws. Also, the possibility of performing the separation of coordinates depends on the coordinate system (it is a privilege of the coordinate system) in which the motion is described. As a simple illustration, consider the motion of a heavy particle on the inner surface of a vertical cylinder, which we considered previously in Section 4.2. Since Hamilton’s function is given by (4.2.45)’ the HamiltonJacobi partial differential equation is of the form
(-)az
a~ + L(57 + 1 as at
2mR2
ae
2m
- mgz
=
0.
(4.4.23)
Since the Hamiltonian does not depend on time t explicitly and 9 is a cyclic coordinate, we seek a complete solution of Equation (4.4.23) of the form
S = -Et
+ A9 + f ( z ) .
(4.4.24)
Substituting this into (4.4.23), we find that
f
= 2m2gz - ( A / R )
+ 2mE.
(4.4.25)
Integrating, we obtain the complete solution of the form S = -Et
+A8 +
+
Applying the Jacobi theorem, it follows that the generalized momenta are given by pe = A , p z = A[2m2gz - (A/R)’ + 2mE]1/2, (4.4.27) and 1/2 as - 8 - 2A _ 2mZgz + 2 m ] = B1 = const., aA 3m2gR2
67 67 +
[
as -- - t _ aE
+
mg
[
2m2gz -
2m]
= B2 =
const. (4.4.28)
According to the given initial conditions, for t = 0 we have
B
= 0,
z = 0,
pe = mRvo cos a ,
pz
=
mvo sin a. (4.4.29)
Therefore, A = mRvo cos a, vo
BZ = -sina, g
2 vt sin a cos a B1 = - ’ 3 SR
mvt E =-
2 .
(4.4.30)
180
4. Motion of Conservative and Nonconservative Dynamical Systems
Using these relations, and eliminating the term [2m2gz - ( L ~ / Z ? ) ~+ 2mE]”z from (4.4.28), we obtain
e = (uom)t cos a Also, from the second equation in (4.4.28), we find that z = gt2 - + uot sin a. 2 Finally, putting this into the expression for p z , given by (4.4.27), we get pz
=
m(gt
+ uosina),
which completes the solution.
4.5. Application of the Hamilton-Jacobi Method to Linear Nonconservative Oscillatory Systems The method of Hamilton and Jacobi is formulated for potential dynamical systems, namely, those whose behavior is completely described by a Lagrangian or Hamiltonian function. As underlined in the introduction of this chapter, many important problems arising in physics, mechanics and engineering have been solved by means of this powerful method. In this section we demonstrate that under some conditions, the HamiltonJacobi method can be applied to a purely nonconservative linear system that does not possess a Lagrangian function. Let us consider a linear dissipative dynamical system whose differential equations of motion are of the form
xi + b.. rjxj’. + w 2Xi
=0
( i , j = 1,
..., n).
(4.5.1)
Conservation laws for this type of dynamical system were studied in Section 3.21. Note that subscripts on the variables are equivalent to the previous superscripts. In order to incorporate gyroscopic forces (whose tensor is antisymmetric: bij = -bji) and dissipative forces (whose tensor is, as a rule, symmetric: bij = bji), we will suppose that the coefficients bij in (4.5.1) are in general not symmetric; that bij # bji. Since the differential equations of motion, (4.5.1), are not derivable from a Lagrangian function, we turn to the possibility that some (combinations) of these equations can be described by means of a Lagrangian function.
4.5. Application of the Hamilton-Jacobi Method
181
Let us take the first differential equation of (4.5.1) as it stands and multiply the second by a constant factor 1 2 , and the third by h3, etc., and then sum all of these equations. Thus, we obtain, for a = 2,3, ...,n,
+ 02(x1+ 2,x,)
(4.5.3)
=0
We introduce a new variable, Y = x1 + Lax,.
Let us select the multipliers 1, (a = 2, 3,
bl2 611
+ 1,b,2
+ 1,bd
= 12,
(4.5.4)
...,n) in such a way that
..., bbll~ n++ Axban = 1,. 1,bd
(4.5.5)
Then, denoting by
2k
=
bll
+ Aab,l,
(4.5.6)
we reduce Equation (4.5.3) to the form Y+2kY+u2Y=Oy
(4.5.7)
which can be derived by means of a Hamiltonian or Lagrangian. By using Equation (4.5.6), we can write the system (4.5.5)in the form of n - 1 homogeneous algebraic equations [see also (3.21.19)]:
If we solve this system with respect 1, and substitute the solution into
182
4. Motion of Conservative and Nonconservative Dynamical Systems
(4.5.6), we obtain an nth-order determinantal equation,
bii - 2k
biz
b21
bzz - 2k
bn i
bn2
.
...
bln
*.*
b2n
=
0,
(4.5.9)
bnn - 2k
as the requirement for a nontrivial solution with respect to 2k. Solving (4.5.9) by supposing that there are no multiple roots, we obtain 2k(l), ...,2k(n). Substituting these into (4.5.8) and solving, we find & ( I ) , & 4 2 ) , ...,.Aa(n) (a = 2,3, . .., n). Therefore, Equations (4.5.4), (4.5.6), and (4.5.7) are in fact
and
qi)= X I + 1 2 a ( ; ) ~ a (i = 1 , ..., n, a = 2,3, ..., n), 2k(i) = bil + L(i)bal,
ci)+ 2k(;)$i) +
o2Ki)=
0.
(4.5.10) (4.5.11) (4.5.12)
In the last equation, and in the subsequent text, the summation convention with respect to the repeated indices in the parentheses ( ) is not assumed, while this convention is applied to all other indices that are not in parentheses. As pointed out already, the system of equations (4.5.12) admits a Lagrangian function of the form '
n
Hamilton's function thus becomes
where the generalized momenta are defined by (4.5.15)
The Hamilton-Jacobi partial differential equation becomes
To solve this equation, we apply the method of separation of variables, that
4.5. Application of the Hamilton-Jacobi Method
183
is, we suppose that
S(t, & I ) ,
..-,Vn))
+ ... + S(n)(t, qn)).
= S(l)(t,ql))+ S(z)(t,q ~ ) )
(4.5.17)
Substituting this into (4.5.16), we get n equations, one for each of the q;,,t ):
S(i)(
For each of the n degrees of freedom, we are led to equations that are completely analogous to (4.3.44). Therefore, by analogy with (4.3.49), we get ~ ( i= )
-
3~ & [ k ( +i ) o tan(ot + ~ ( i ) ) ] e ~ ~ ( ~ ) '(4.5.19) ,
where the C(i)[ ( i ) = 1, ...,n] are constants. Applying the Jacobi theorem, we have
and
From this last group of equations, it follows that
qi)= [-
~B(~)/CO]'/~~-'(I'' COS(CO~ + C(i)).
(4.5.22)
Using (4.5.22), the generalized momenta, (4.5.20), can be expressed as functions of time: = [k(i)(-2B(i)/o)'''
COS(CO~ + C(i,) + (- 2B(i)a)'/' sin(ot
+ C(i))]ek(O'. (4.5.23)
Let us suppose that the initial conditions for the system (4.5.1) are given as bi. Using (4.5.10) and (4.5.15), we can find the initial values qi)(O) andpi(0). Therefore, from (4.5.22) and (4.5.23), we can find the 2n constants B(i) and C(i)in terms of ai and bi. The motion of the dynamical system is given by (4.5.10) and we suppose that this expression can be inverted to give Xi = xi(t). The method of Hamilton-Jacobi can also be applied to dynamical problems whose differential equations of motion are of the form
x;(O) = ai and Xi(0) =
2i
+ 2kki + aijxj = 0
(i, j = 1, ...,n).
(4.5.24)
Conservation laws for this type of dynamical system are discussed in Section (3.20).
184
4. Motion of Conservative and Nonconservative Dynamical Systems
In order to include nonconservative position forces into consideration, we suppose that the coefficients aij are not symmetric: a0 # aji . Therefore, the dynamical system is not conservative and a Lagrangian function cannot necessarily be found. However, repeating the same steps as in the previous case, namely, multiplying the second equation (4.5.24) by a constant factor p2, the third by p3 etc., and summing all equations, including the first, we obtain, for a = 2 , 3 ,..., n,
+
where X
= x1
Denoting by
+ (sin + p,aan)Xn = 0,
*.*
+ pax,
(a!
2,3,
=
(4.5.25)
..., n).
(4.5.26) (4.5.27)
and selecting p, in such a way that a12 all
+p
+ p,a,l
~ -2 - p2,
we reduce (4.5.25) to
ain
+ paaan = p n ,
..., all + p,a,l
+ 2 k Z + 0 2 X = 0.
X
To determine o2and the unknown multipliers p, use (4.5.27) to express the system (4.5.28) as
+
pz(a22 - 0 2 ) p3 a32
+
+
~ 2 a 2 ~ ~ 3 a 3 ~
(4.5.29) (a!
+ - - - + pnan2
+ pn(ann
(4.5.28)
=
=
- 02) =
2, 3 ...,n), we first
- a12
-aln.
Solving this system with respect to p,, we substitute the result into (4.5.27) and, after a simple transformation, we obtain the determinantal equation all
- o2 a21
an1
ain
a12
a22
azn
- o2
an2
(Inn
-
= 0,
(4.5.31)
o2
which is an algebraic equation of the nth order with respect to
02.
4.5. Application of the Hamilton-Jacobi Method
185
Therefore, from (4.5.31) and (4.5.30), we find o t l ) , ..., o t n ) and ...,p,(n) [assuming that (4.5.31) has distinct roots]. From (4.5.26), (4.5.27), and (4.5.29), we actually have
p,(l),
X(i)
= XI
+ p a ( i ) X , (a = 2, 3, ...,n, (i) = 1, ..., n), 4) = a l l + p,(i)a,i, 2 ( i ) + 2 k ~ ( i+ ) o h ) X ( i ) = 0,
(4.5.32) (4.5.33) (4.5.34)
where the repeated indices in the round brackets (i) are not summed. Since the system of transformed differential equations (4.5.34) is derivable from a Lagrangian function of the form n
L =
C +[x& - W $ ) X , $ ) I ~ ~ ~ ~ , i=l
(4.5.35)
we can, repeating the same procedure as outlined in the previous case, apply the Hamilton-Jacobi method. For purely conservative dynamical systems whose Lagrangian function is L
= La..%. 2 IJ IXJ .. - 1 ~CijXiXj,
(4.5.36)
where a i j = a j i and Cij = C j i , the method suggested in this section, which is based on the introduction of undetermined multipliers A,, is identical to the method of principal (normal) coordinates in linear vibration theory. However, the method described here, in contrast to the method of normal coordinates, does not require the knowledge of any Lagrangian function and can be applied, as we demonstrated, to nonconservative cases that do not possess Lagrangian functions. Frequently, we can apply the method of multipliers to forced vibration problems, which will be demonstrated by means of the following example. 4.5.1. The Linear Forced Vibration System with Two Degrees of Freedom As a practical example, one that serves as an illustration of the foregoing general considerations, let us study the motion of a linear forced system with two degrees of freedom, whose differential equations of motion are of the form a l l x l + ~ 1 2 x 2+ c l l x l + ~ 1 2 x 2= H I sin(pt + 4, (4.5.37) ~ 1 2 x 1+ a 2 2 A + ~ 1 2 x 1+ ~ 2 2 x 2= Hzsin(pt + a),
where a i j = parameters.
aji,
Cij = C j i ,
and H I , H2, p, and 6 are given constant
186
4. Motion of Conservative and Nonconservative Dynamical Systems
Multiplying the second equation by a constant multiplier A and adding it to the first, we find that (all
+ A a d X 1 + (a12 + Aa22)x2 + (c11 + Ac12)xl
+ ( c 1 2 + A c 2 2 ) x z = ( H I + AHz)sin(pt + 6)
(4.5.38)
or (a12
+ Aa22)
all a12
+ Aa12 + La22
c11 c12
= (HI
Dividing by
a12
+ Ac12
+ Acz2 x 1 + x 2
+ AH2)sin(pt + 6)
(4.5.39)
+ A a 2 2 and denoting by (4.5.40)
it is clear that under the condition (4.5.41)
we can write the differential equation (4.5.39) in the form of the following system of differential equations :
where the repeated (i) is not summed and quadratic equation A2 -
a22c11 a12c22
- a11c22 A+ - a22c12
A(1)
and
=
+ A ( i ) a i 2 x1 + x 2 a12 + A(i)a22 all
= K(i)Xl
+ xz
are the roots of the
a11c12
-
~ 1 2 ~ 1 1
a12c22
-
a22c12
which follows from (4.5.41). In Equation (4.5.42), X(i)
4 2 )
=
=
0,
X ( i ) denotes
(4.5.43)
the quantity
+ x1 + x2 1 + 2 A(i)~22
~ 1 1 A(ilc12 ~
[(i) = 1,2].
(4.5.44)
Finding A 1 and A 2 from (4.5.43), we also find the squares of circular frequencies [see (4.5.40)] : (4.5.45)
4.5. Application of the Hamilton-Jacobi Method
187
Before we apply the Hamilton-Jacobi method, it is of interest to find a more explicit form of the parameter K , which was introduced in Equation (4.5.41).According to (4.5.41),A(i) and K(i) are connected in the following way:
From these equations, we find that - a11c22 - K(i) = a22c11 .
A(i)
-K(. A . ’)
=
(I)
a12c22
- a22c12
allCl2
- alZCll
a12c22
- a22c12
Y
(4.5.47)
-
However, from these expressions, it follows that for A(1)
=
- K(z)
and
A(z) = -Kw,
(4.5.48)
we have A(1)
+ 42)
A(l)A(2)
= =
a22c11
- a11c22
a12c22
- a22c12
UllC12
- U12Cll
a12c22
- azzc12
9
(4.5.49)
Hence A(1) and A( 2) are the roots of the quadratic equation (4.5.43). In order to incorporate the differential equations (4.5.42) into a Lagrangian problem suitable for application of the Hamilton-Jacobi method, we try to find a Lagrangian function of the form
L =
i[z&- Alpcos(pt + 41’
+ 41’ + +[k&- A2p cos(pt + 41’ - +[ X( Z)- A Zsin(pt + a)]’, -
o&[X(1)- A1 sin(pt
(4.5.50)
where A1 and A Z are constant parameters to be determined in such a way that the Lagrange equations,
(4.5.51) generate the correct differential equations, (4.5.42). Substituting (4.5.50) into (4.5.51),we obtain
188
4. Motion of Conservative and Nonconservative Dynamical Systems
Comparing these equations with (4.5.42), we find that A1 =
Hi
(a12
+ A1H2
+ Alazz)(w?l) - p2)'
(4.5.53)
Hi + A 2 H 2 A2 = (a12 + A2(122)(0?2)- P2) * Introducing the generalized momenta by the relations
(4.5.54)
the Hamiltonian function is found to be H = '2 2 P1
+ AlPlpCOS(pt + 6) + AzPzpcos(pt + s)
+
+ &?i)[X(1)
- A1 sin(pt
+ 6)12 + +o&[X(2)- A2 sin(pt + )I2
(4.5.55)
Therefore the Hamilton-Jacobi partial differential equation is
+- as
A2p cos(pt
aX(2)
+ s)+ +o&[X(1) - A1 sin(pt + @I2
+ +o?z)[X(~) - A2 sin(pt + s)I2
= 0.
(4.5.56)
We seek a complete solution of this partial differential equation, of the form S = ~ [ X ( I-) A1 sin(pt
+ i5)I2f1(t) + +[X(Z)- A2 sin(pt + 6)12f2(t),
(4.5.57)
where fl(t) and f 4 t ) are unknown functions of time. Substituting (4.5.57) into (4.5.56), we obtain f1 + fi" + ot1) = 0 and f2 + fi" + o("z) = 0. Integrating, we find f1 = - o(1)tan(o(1)t + Cl) and f2 = -w(2) tan(o(2)t + C2), where C1 and C2 are arbitrary constants. Therefore, the complete solution of (4.5.56) is S = -&o(l)[X(l) - A1 sin(pt
-+[X(2) - A2sin(pt
+ S)I2tan(o(1)t + CI)
+ ~3)]~0(2) tan(o(2)t + C2).
(4.5.58)
189
4.5. Application of the Hamilton-Jacobi Method
Applying the Jacobi theorem, we get the generalized momenta:
as
pl=--
ax,,
-
-o(l)[X(l) - A1 sin(pt
+ 4 1 tan(w(1)t +
as p 2 = - - - -O(Z)[X(Z) - A Z sin(pt + S)l tan(o(z)t +
Cl),
(4.5.59) CZ),
and
where B1 and BZ are arbitrary constants. The last two equations can be solved with respect to the normal coordinates
+ Cl) + A1 sin(pt + 6), X ( Z )= (- ~Bz/w(z))'/~ cos(w(gt + CZ)+ A Zsin(pt + a), X(1) = (- 2B1/0(1))'/~ cos(o(1)t
(4.5.61)
where A1 and A2 are given by (4.5.53). However, in accordance with (4.5.44) and (4.5.48), X(1) = -A(z)X1 4 2 )
Solving for XI and
XZ,
= -A(l)Xl
+ x29 + x2
(4.5.62)
*
we have
Substituting (4.5.61) into (4.5.63), it follows that
+ Cl) + DZcos(o(~)t+ CZ)+ R1 sin@ + 6), xz = 1(1)D1cos(o(1)t + Cl) + L(Z)DZ cos(w(2)t + C Z )+ RZsin(pt + 6) XI
= D1 cos(o(~)t
(4.5.64)
where
190
4. Motion of Conservative and Nonconservative Dynamical Systems
(4.5.65)
-
-
P2)
42)(Hl (a12
+ L(Z,H2)
+ 42)a22)(0?2) - P2)
Note that using (4.5.49) for A = A(1) and 1 = 4 2 ) , as well as (4.5.48), (4.5.49, and (4.5.46), we can (after laborious but simple algebraic manipulations) transform the last two expressions to the following form: R1 = R2 =
- U 2 2 P 2 ) - H2(c12 - U 1 2 P 2 ) (a11a22 - a?2)(0?1,- P2)(0?2) - p 2 )’ Hl(C22
a11p2) - H l ( C l 2 - a12p2) - a?2)(~?1) - p2)(wt2)- p 2 ) ’
HZ(Cl1 (ma22
(4.5.66)
4.6. A Field Method for Nonconservative Dynamical Systems
As briefly demonstrated in the previous sections, the Hamilton-Jacobi method can be advantageously used in many practical situations as an exact method for solving the canonical differential equations of motion. In addition, a variety of approximate procedures can be built up, based on this method, for solving the nonlinear problems for which an exact, complete solution of the Hamilton-Jacobi partial differential equation is not available. For example, the interested reader will find an exhaustive review of applications of the Hamilton-Jacobi method within the realm of nonlinear vibration theory and celestial mechanics in the monographs of Kevorkian and Cole [lo] and Nayfeh [ll]. However, the method of Hamilton and Jacobi can be applied only to those systems that are completely described by the LagrangiadHamiltonian function, and the purely nonconservative dynamical systems stay outside of the areas treated by this method. In this and the following sections of this chapter, we discuss a field method that is suitable for studying the motion of purely nonconservative dynamical systems. The method presented here can be also applied to conservative dynamical systems and to those nonconservative systems that are
4.6. A Field Method for Nonconservative Dynamics1 Systems
191
generated from a LagrangiadHamiltonian. In this respect, the field method presented here differs completely from the Hamilton-Jacobi method. Let us consider a nonconservative, holonomic, rheonomic, dynamical system whose differential equations of motion are of the form fl
=
Xl(t, x1,
...,xn), (4.6.1)
fn
= Xn(t, x1,
...,x n ) .
Note, that by the term “nonconservative,” we understand that the number n may not be an even number; also, we do not insist that we be able to write the system (4.6.1) in the canonical form (4.2.20). The basic supposition of our theory is based on the hypothesis that one coordinate of the system, say X I ,can be selected as the basicfield, depending on time t and the rest of the coordinates describing the dynamical system: x1
=
cp(t,x2,
...,xn).
(4.6.2)
Differentiating this expression totally with respect to time and using the last n - 1 equations of (4.6.1), we write the first differential equation of this system in the form acp
at
acp + -X,(t, ax, = O
cp,x2,
...,xn) - Xl(t, cp,x2, ...,x n )
( a = 2 , 3 ,..., n),
(4.6.3)
where the summation convention with respect to repeated indices is applied. We call the partial differential equation (4.6.3) the basic field equation. It plays the same role in our considerations as the Hamilton-Jacobi partial differential equation in the previous discussion. However, it should be noted that, in contrast to the Hamilton-Jacobi partial differential equation, which is, as a rule, strictly nonlinear, the basic field equation (4.6.3) is a quasilinear partial differential equation and, therefore, has a simpler structure than the Hamilton-Jacobi equation. That is to say, the solution of the basic field equation (4.6.3) forms a hypersurface in the n-dimensional space t, x 2 , ...,xn . According to the geometric theory of quasi-linear partial differential equations (see, for example, [l]), this hypersurface is generated by the family of characteristic curves whose differential equations are given by the system of ordinary differential equations (4.6.1). In general, the geometric properties of the integral surface O(t,xz , ...,xn)are considerably simpler than the geometrical properties of the integral surface S(t,X I , ...,xn) of the corresponding Hamilton-Jacobi theory, since the gradients aS/ax’ appear nonlinearly in the Hamilton-Jacobi partial differential equation.
192
4. Motion of Conservative and Nonconservative Dynamical Systems
In order to demonstrate certain variational properties of the basic field equation, we consider the differential equations of motion in the nonconservative Hamiltonian form (4.2.17), namely,
where x i (i = 1, ...,n) are the generalized coordinates and pi are the generalized momenta. The Qi are purely nonconservative forces. As shown in Section 4.2, the equations of motion can be related to the central Lagrangian equation, (4.2.12), which under the supposition that 6d - d6 = 0, has the form d -(pi 6 ~ '-) 6(piki - H ) - Qi 6 ~ =' 0 dt
(i = 1, ..., n). (4.6.5)
By adding and subtracting the time derivative of the quantity Udx' in the last equation, where U is an arbitrary function depending on time t , on the generalized coordinates x l , ..., xn, and on the n - 1 components of the momentum vector p2 = ZZ,...,pn = Z n , we find that, for i = 1, ...,n, d d z ( p i S X i - U ~ X '-) 6(pi6Xi - H ) + Qi6Xi + - ( U ~ X ' )= 0. (4.6.8) dt
At this point, we define the first component of the momentum vector p1 to be equal to the field function U : p1 = U(t,X1,...,~
...,
" ~ 2 2 , zn).
(4.6.7)
Therefore, we can write (4.6.6) in the form d dt
-(z, SX,) - 6(UX'
d + z,Xm - H ) - Qi 6 ~ -' Q,6 ~ +" dt - ( U ~ X ' )= 0
(4.6.8)
for 01 = 2,3, ..., n. Writing this equation explicitly, and taking into account that dx', ax", and 62, are independent, we obtain the following system of equations :
(4.6.9)
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
193
Calculating i, and i afrom the last two expressions and inserting them into the first equation of the system (4.6.9), we find the following basic field equation for the system (4.6.4):
=0
(a! =
2 , 3 , ...,n)
(4.6.10)
in which, according to (4.6.7), the basic field is taken to be the first component of the momentum vector. As indicated previously, we can derive the basic field equation (4.6.10) by substituting the field (4.6.7) into the equation p1 = - ( a H / a x l )
+ Q~
(4.6.11)
and using all of the rest of the equations of the system (4.6.4). However, the derivation given here is of mechanical significance, since it reveals the invariant character of the basic field equation, which is based on the central Lagrangian equation, i.e., D'Alembert's principle. The field approach proposed here was introduced in References [ 121 through [ 141. Naturally, the set of differential equations (4.6.1) and (4.6.4) are fully equivalent for the majority of the dynamical systems. However, in the text that follows, we will consider only one type of system, without any loss of generality.
4.7. The Complete Solutions of the Basic Field Equations and Their Properties In what follows, we shall be concerned with the complete solutions of the basic partial differential equation (4.6.10). We define a complete solution to be a relation of the form p1 =
U(t,X1,..., ~
...,
" ~ 2 2 , ~ n C1, ,
..., Czn);
(4.7.1)
in addition to the arguments t , x', ..., x", 2 2 , ..., z,, , the solution p1 contains 2n arbitrary constants and, when substituted into Equation (4.6.lo), reduces it to an identity. Note that every complete solution satisfies the property mentioned in Section 4.3; that is, if we eliminate the 2n constant parameters C1,...,Czn from the 2n + 1 relations (4.7.1), aU/at, aU/axi, and aU/az, ( i = 1, ..., n, a! = 2 , 3 , ..., n), we should obtain the basic field equation (4.6.10).
194
4. Motion of Conservative and Nonconservative Dynamical Systems
4.7.1. Bundles of Conservation Laws Several observations can be made. With respect to the differential equations of motion (4.6.4), the expression (4.7.1) represents a bundle of conservation laws for the dynamical system. That is to say, we can recover all of the 2n conservation laws of the dynamical system (4.6.4) from the complete solution (4.7.1) of the basic field equation by giving any particular values to the 2n - 1 constants Ci, in their relevant domain, and by allowing one of them to be arbitrary. For example, the set of relations
p1 = Ul(t,X1,..., xfl,22,..., Z n , cl),
p1
=
u2n(t,x1,..., x ~ , z..., ~ ,~
=
~2
~1
n c Z, ~ ) ,
~3
=
=
= CZ =
~ 2 = ,
0.
= ~ 2 n - 1=
0.
represents a complete set of conservation laws for the dynamical system (4.6.4) under the condition that all of these relations are mutually independent. In other words, under the condition that aUi/aCi # 0 (i = 1,2, ...,2n), we can invert (4.7.2) to obtain the more familiar form for the conservation laws:
@l(t,x l , ..., x n ,21, @2n(t,x l , ...,X ' ,
..., 2,)
=
c1, (4.7.3)
21
. . a ,
2,) = ~ 2 n
(zi
pi).
4.7.2. Initial Conditions and Conditioned Forms The bundle of conservation laws (4.7.1) can also be matched with the initial conditions of the dynamical system, if they are formulated as an initialvalue problem. Let the initial conditions of the system (4.6.4) be given by
~'(0) = a',
pi(O) = bi,
(4.7.4)
where a' and bi are given constants. Applying (4.7.4) to (4.7.1) and expressing one constant, say CI,in terms of a', bi, and the rest of the parameters C2, ..., C z n , we can write (4.7.1) in the following conditioned form: p1 = O(t,x', ..., x n , 2 2 , ...,Z,, a1, ...,an,b l , ...,b,, c2, ..., C2,).
(4.7.5)
It should be noted that it is a characteristic feature of the theory that the constant parameters C2, ..., C2n in the conditioned-form solution, (4.7.9, remain undetermined through the entire process of finding the motion.
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
195
We will now demonstrate that the motion of the dynamical system, which moves in accordance with the differential equations of motion (4.6.4) and the initial conditions (4.7.4), can be obtained from the expression (4.7.5) and the 2n - 1 algebraic equations
-a -0 - 0
( A = 2 , 3 , ..., 2n)
acA
(4.7.6)
for arbitrary values of the constants CA. We suppose that the following functional determinant:
a20 ... a2O a20 ... ___ ac2ax1 ac2axn ac2az2
~
~
a20 ~
aC2
az, (4.7.7)
a20
ac2,ax'
...
a20
a20
aC2, ax"
aC2,az2
...
a20 aC2,
az,
is nowhere zero in the relevant domain of xi, z, and CA (i = 1, ..., n, (Y = 2 , 3 , ..., n, A = 2, 3 , ..., 2n). It is clear that the requirement (4.7.7) can be interpreted as the condition of solvability of the 2n - 1 Equations (4.7.6) with respect to the 2n - 1 quantities xi and zol.However, we will suppose that Equations (4.7.6) are not solvable with respect to the CA, that is, they become an identity for t = 0, for arbitrary values of CA. To prove the previous statement, we first differentiate Equation (4.7.6) totally with respect to time:
a20 x i + a2ii 2, +acAat acAa2 acAaz, a20
~
( A = 2 , 3 , ..., 2n, i
=
1,2,
(4.7.8)
=0
...,n, a = 2 , 3 , ..., n).
On the other hand, since the right-hand side of (4.7.5) satisfies the basic field equation (4.6.10) identically, we can form a partial derivative of the basic field equation (4.6.10) with respect to CA, to obtain.
a20
a20 a H
a20 a H
+
a2H [-- a 0 2 +--axwaz,ao + aOa2H
ax'
aO
az,
a20
aH
L au (n.- g) (4.7.9)
196
4. Motion of Conservative and Nonconservative Dynamical Systems
Employing (4.7.6) and forming the difference between (4.7.8) and (4.7.9), we obtain
+ 5(2, +
acAaz,
(A = 2 , 3 ,..., 2 n , a = 2 , 3 ,..., n). This is a set of linear algebraic equations that is homogeneous in the quantities
Since the determinant (4.7.7) is different from zero by hypothesis, it follows that . aH aH X' = ( a = 2 , 3 ,..., n , i = l , ..., n) a n ' i , = p , = - - +ax.Q , (4.7.1 1) which are the 2n = 1 canonical equations (4.6.4). In order to show that the first canonical equation,
aH
P I = --
ax' +
(4.7.12)
Qi,
is also satisfied, we form the time derivative of (4.7.5) and combine the resulting equation with (4.7.11):
au
auaH ax, az,
. auaH pi = - + - - +--+-
at
ax'au
au
az,
( ax..
aH
)
--+Qa.
Substituting this into the basic field equation (4.6.10), we find that Equation (4.7.12) is now satisfied, which completes the proof. 4.7.3. General Solutions
Finally, we demonstrate how to obtain the general solution of the nonconservative canonical system (4.6.4) from a complete solution of the basic field equation, (4.7.1). Suppose that one of the constants in the complete solution, say C1, can be considered to be an arbitrary function of all other constants
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
197
CZ,.-.,C 2 n ; C1
= Cl(C2, ..., C 2 n ) .
(4.7.13)
In what follows, we will show that a complete solution of the basic field equation (4.7.1), together with the 2n - 1 algebraic equations
au au acA ac,
-+ -DA
=
( A = 2, 3,
0
..., 2n),
(4.7.14)
defines a family of general solutions of the nonconservative canonical system (4.6.4), where
ac, acA
( A = 2 , 3,..., n)
DA = -,
(4.7.15)
is a set of new constant parameters. To prove this, we form the time derivative of (4.7.14):
a2u a2u . a2u +acAat acAaxi2 + acAaz, 2, a2u . X I. + -2.1 a2u ac, at +-ac, ax1 ac, az,
= 0.
(4.7.16)
Since, for a complete solution of the form (4.7. l), the basic field equation (4.6.10) is an identity, we can differentiate partially with respect to CA and obtain
-(? a 2 u
a2u +- a 2 u DA + aH acAat at ac, au
ax
+ -DA) a2u acA axlac,
a2u a2u
a u + ac, au (-acA
-DA)
=
0.
(4.7.17)
Employing (4.7.14), and taking the difference between Equations (4.7.16)
198
4. Motion of Conservative and Nonconservative Dynamical Systems
and (4.7.17), we find that
a2u
a2u
a2u
(4.7.18)
Therefore, under the supposition that the following determinant is different from zero: a2u
\
a2u
det( det
a 2 u + - 0 2a 2 u +-D2 ac2 axn ac, ax'
a2u +- a 2 u D2 ... -+ac2. az. ac, az. a c 2 az2 ac, az2
j
a2u a2u ac2. ax1 +-Din a c l ax1
a2u ... -+-D2n ac2. ax'
a2u ac, axn
a2u a2u a2u a2u Din ... -i-D2n ac2, az2 ac, az2 ac2. az, ac, az,
-+-
I
(4.7.19)
it follows that the 2n - 1 canonical equations ,
aH api
= -,
pa=--
aH ax. + Q ,
( i = 1 , ..., n , a ! = 2,..., n) (4.7.20)
are satisfied for arbitrary values of the constant parameters C1, ..., Czn, D2, ...,Dzn . To show that the first equation, p l = -(aH/axl) QI, is satisfied also, we form the time derivative of (4.7.1) and combine the resulting equation with (4.7.20):
+
Substituting this last expression into the basic field equation (4.6. lo), it follows that the first canonical equation is also satisfied for the arbitrary values of the C's and D's, which completes our proof. It follows from our analysis that Equations (4.7.1) and (4.7.14) contain 4n - 1 arbitrary constant parameters C1, ..., Czn and Dz, ..., D2n. Consequently, we have 2n - 1 superfluous constants remaining arbitrary during the process of finding the motion of the nonconservative dynamical system (4.6.4). The method for getting a general solution of (4.6.4), which is free of this redundancy, is contained in the following paragraphs.
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
199
Let us suppose that we have found a complete solution of the basic field equation (4.6.10) in the form of (4.7.1): p1 = U ( t , x l , . . . , x " , z ~. -, . , ~ n , ~
1
,
~2,).
(4.7.21)
Solving this equation with respect to CI, we find a relation of the form
.. .,x", U,2 2 , ...,Z n , C2, ..., C2n). (4.7.22) = 2 , 3 , ...,2n)enter linearly into (4.7.14), we can
C1 = '€'1(t,xl,
Since the constants DA (A solve these algebraic equations with respect to DA and then substitute (4.7.22) into the resulting expressions. The resulting equations are of the form D2 = Y2(t,XI,...,x", U,2 2 ,
...,Zn, C2, -..,C 2 n ) , (4.7.23)
D2n = Y2n(t,X I ,
...,x", U,2 2 , ...,Zn, C2, ..., C2n).
The relations (4.7.22) and (4.7.23) form a family of general solutions of the differential equations of motion (4.6.4) in which the parameters CA (A = 2,3, ..., 2n) are arbitrary. Therefore we are free to choose any particular value for these parameters in their relevant domain. Hence by putting U = p1, 2 2 = p2 , ...,Z n = Pn ; we have ~1
=
~ ( xl, t , - -. xn,
.- -
pn), (4.7.24)
D2n = Ylzn(f,xl,. . . , x " , P I ,. . . , p n ) .
The procedure described here for finding the family of general solutions of the nonconservative canonical equations (4.6.4) was introduced in Reference [13]. In order to clarify the previous discussion, we turn 'to an example. Let us consider the nonconservative dynamical system whose differential equations of motion are of the form 21 = X 2 ,
Xl(1) = a,
(see [15]). Taking as the field variable xz
=
U(t,XI), we find the basic field
200
4. Motion of Conservative and Nonconservative Dynamical Systems
equation in the form
-au + at
u2+ -u= 0. u-au - ax,
x1
t
(4.7.26)
The variables XI and t are separated by the assumption that (4.7.27) from which we find that (4.7.28) where an overdot denotes time differentiation and a prime denotes differentiation with respect to X I . This expression can be split into the following system:
Integrating, we find a complete solution of the basic field equation (4.7.26) of the form XI lnxl + A (4.7.29) x2 = t lnt + B ’ where A and B are constants. We now discuss the various possibilities of employing this complete solution in the same order as the observations given in Sections 4.7.1,4.7.2 and 4.7.3. i. As indicated in 4.7.1, we can interpret (4.7.29) as a bundle of conservation laws of the differential equations of motion (4.7.25). For example, taking in (4.7.25)’ A # 0, B = 0 and A = 0, B # 0, we find the following two independent conservation laws of the dynamical system (4.7.25) x2
-tlnt
XI
XI
-
- lnxl
x2 t
lnxl = A = const.
(for B = 0) (4.7.30)
-
In t = B = const.
(for A = 0).
Note that A and B can assume arbitrary values in these two conservation laws for (4.7.25). Due to this fact, we call the complete solution (4.7.29) a bundle of conservation laws for the dynamical system (4.7.25).
201
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
ii. To find the motion that corresponds to the given initial conditions = a, x2(1) = b, we substitute these into (4.7.29) and calculate A = B(b/a) - lna. Putting this into (4.7.29), we obtain the conditioned form solution of the basic field equation, which is matched with the initial conditions xl(1)
xz = &t,
XI,
ln(xl/a) + B(b/a) t lnt + B
x1
a, b, B ) = -
(4.7.3 1)
where B is completely arbitrary. According to (4.7.6), we find the motion form (4.7.31) and also
a0
-=
aB
0.
(4.7.32)
It is easy to verify that, under the condition (In t equation gives XI = at@/@.
+ B)2 # 0 ,
the last (4.7.33)
To find the second coordinate as a function of time, we substitute (4.7.33) into (4.7.31). The parameter B disappears and we find x2 = bt(b/@-l 9 (4.7.34) which completes the solution of the initial value problem. iii. To find the general solution of (4.7.25) from the complete solution (4.7.29), we take A = A ( B ) . Hence, the equation
-axZ =o
(4.7.3 5)
aB
is equivalent to lnxl = -(lnt dA dB
+ B) - A
(4.7.3 6 )
By substituting this into (4.7.31), we find (4.7.37)
and (4.7.36) gives lnxl = D1lnt
+ D2
(D2
=B
g
-A).
(4.7.38)
The last two equations represent the general solution of the equations of motion (4.7.25).
202
4. Motion of Conservative and Nonconservative Dynamical Systems
4.8. The Single Solutions of the Basic Field Equation If we are able to find a solution of the basic field equation (4.6.10) that depends on a single constant C1, p1
=
U(t,xl,
-.. x", 2 2 ...,Z n , cl),
(4.8.1)
then it is clear that it represents a conservation law of the dynamical system (4.6.4). Under the condition aU/aC1 # 0, we can invert (4.8.1) to obtain the more familiar form of the conservation law: Q(t,xl,
..., x n , p l ,...,p n )
=
~
1
.
(4.8.2)
Naturally, if we are able to find 2n single solutions of the basic field equation (4.6.10), Q(t,xl,
(i = 1, ...,2n),
...,x", p i , .. .,Pn) = Ci
(4.8.3)
then the problem of finding the solution of the dynamical system (4.6.4) is solved, provided that the set of conservation laws (4.8.3) is mutually independent. That is equivalent to the requirement that the Jacobian be different from zero: det( xl,
C1,
ae.9
C2n
...,x n , P I ,
) #O.
.-.,pn
(4.8.4)
In practical applications of the field method discussed here, we are frequently able to find all of the single conservation laws more easily than a complete solution of the basic field equation.
4.9. Illustrative Examples
In this section, as an illustration of the foregoing theory, we will discuss several dynamics problems frequently arising in practical applications.
4.9.1. The Linear Damped Oscillator Consider the motion of the linear damped oscillator, whose differential equations of motion are Xi
= xz ,
X2
= -w'x~ -
2kx2,
(4.9.1)
where w and k (w > k) are given constants. Taking as the field variable
4.9. Illustrative Examples
203
x1 = U(t,xz),the basic field equation is of the form
au - (o’U + 2 k ~ 2au _ ) - - xz = 0. at ax2
(4.9.2)
In order to obtain single solutions of the basic equation (4.9.2), we suppose that they are of the form
x1 = U(t,x2) = A X 2
+f(t),
(4.9.3)
where A is an unknown constant and f ( t ) is a function of time (see [16]). Substituting these into (4.9.2) and equating the free terms and the terms containing xz to zero, we obtain, respectively,
f- d A f and (02A)’
=0
+ 2k(w2A)+ 0’= 0.
(4.9.4) (4.9.5)
From the last equation we find the following two values for A : (u2A)1= - k
+ ni,
(02A)2= - k - ni,
(4.9.6)
where n = (m2 - k2)’/’,
j =
(4.9.7)
By integrating (4.9.4), we find the following two single solutions of the basic field equation (4.9.2):
which represent, at the same time, two independent conservation laws of the dynamical system (4.9.1). It is interesting to note that the product C1C2 will yield the quadratic conservation law of energy type, considered previously (see Section 3.8): (4.9.9) (xi + 2kx1x2 + cu2x?)ezkf= const.
To obtain a completesolution of the basic field equation (4.9.1), we seek one with two unknown functions of time: XI =
U ( t , ~ z=) Fi(t)xz + Fz(t).
(4.9.10)
Inserting this into (4.9.2), we arrive at the following system: F1
-
w 2 F f - 2kF1 - 1
=
0,
F z - CO’F~FZ= 0.
(4.9.11)
204
4. Motion of Conservative and Noneonservative Dynamical Systems
The solution of this system is found to be Fl = [-k
+ n tan(nt + D l )01l ~ , (4.9.12)
Fz
=
D2e-k'
cos(nt
+ Dl) '
where D1 and D2 are constants. Thus, we find that a complete solution of the basic field equation (4.9.2) is
x1=
k
x2(-2
+
5)
tan(nt
DZe -kt + 01) + cos(nt + 01)
'
(4.9.13)
According to the discussion in Section 4.7, this expression represents a bundle of conservation laws of the linear damped oscillator (4.9.1). Putting, for example, D1 # 0, DZ= 0 and D1 = 0, DZ# 0, we obtain two independent conservation laws for (4.9.1):
x2
=x
k
l ( - z
+
5)
(4.9.14)
Dze-k' tan(nt) + cos(nt)
or, in inverted form,
-nt
(Di = O),
~
+ arctan
(2+ 3 -
-
=
D1, (4.9.15)
Finally, let us suppose that the dynamical problem (4.9.1) is formulated as an initial value problem: xl(0) = al,
x2(0) = az.
(4.9.16)
Substituting these into the complete solution (4.9.13) and calculating DZ, we find that
4.9. Illustrative Examples
205
Inserting this expression into (4.9.13), we obtain the following conditionedform solution :
)]cos ~1 - (na2/02)sin ~1 + Ial + (k u d o 2cos(nt + Dl)
e -kt . (4.9.17)
To obtain the solution of the initial-value problem (4.9.1) and (4.9.16), we form, in accordance to the rule (4.7.6), the expression
au
-=
aD1
0,
which yields the equation of motion
1
+ k ~ sin(nt) 2 n
e-kf.
(4.9.18)
In order to obtain the second equation of motion, we can, according to the theorem cited in Section 4.7.2, choose any arbitrary value for the constant D1 in (4.9.17). For example, taking D1 = 0, this expression becomes
=
x2
(-k
+ n) tan(nt) + a1 + (ka2/02)e -kt cos(nt)
o2
(4.9.19)
However, instead of selecting any particular value for D1 in the conditionedform solution (4.9.17), we can calculate this expression along the trajectory xz = x2(t) that is given by (4.9.18). Therefore, substituting (4.9.18) into (4.9.17), the parameter D I completely disappears and, after some elementary calculations, we obtain the more familiar form of the solution:
x1 =
x2 a1, a 2 , m Y
a1 cos(nt)
IX2
=R(f)
1
+ -n1 (alk + a2) sin(nt)
e-kt
(4.9.20)
206
4. Motion of Conservative and Nonconservative Dynamical Systems
4.9.2. Projectile Motion with Linear Air Resistance
Consider the motion of a heavy particle moving in a vertical plane with linear friction depending on the velocity. If x and y denote the horizontal and vertical axes, respectively, the differential equations of motion are x
pi
3
= P1, =
-kpi ,
p2
= p2 = =
z,
(4.9.21)
-kz - g ,
where k and g are given constants. Let the initial conditions be x(0) = 0 ,
p1(0) = Uo cos a,
y(0) = 0 ,
z(0) = m s i n a ,
(4.9.22)
where uo is the initial velocity of the particle and a is the initial angle of inclination. Taking as the field variable the momentum p1 = U(t,x, y , z), we arrive at the basic field equation
au
au
au
au
(4.9.23)
-+U--+~--(kz+g)-+kU=O. at ay az ax
+
Seeking a complete solution of the form U = f l ( t ) x + fz(t)y + f3(t)z f4(t) and repeating the procedure similar to the previous example, a complete solution of (4.9.23) is found in the form P1 = Ul(t,x , y , z, c1, ..., C4) -
k C ~ x+ CZY+ [ ( C d k )+ C3ek']z + (gCz/k)t + (gC3/k)ek' +
ekt -
~1
C4
(4.9.24)
Applying the initial conditions and expressing the constant C4 in terms of C1, CZand C3, we have the conditioned momentum:
1
+ ~ ~ c o s a -( 1CI) +
(4.9.25) Let us form the algebraic equations
-a=0o ,
acl
-a -0
-
ac2
0,
arSi
- = 0.
ac3
(4.9.26)
4.9. Illustrative Examples
207
However, after we finish the partial differentiation then, according to the theorem cited in Section 4.7.2, we may choose any particular values for C1, CZand C3. Taking, for example, CI = C2 = C3 = 0 in (4.9.26), we find, respectively, 1 x = - vo cos a(l - e kf ),
k
y
+ -k1z
+ -uok- h a ,
g
= --t
k
(4.9.27)
For the same particular values of C1, CZand C3, we obtain, from (4.9.25), p1
=
(vO
cos
(4.9.28)
which completes the calculation of the motion of the system. 4.9.3. A Nonconservative Coupled System Consider the following nonconservative dynamical system with two degress of freedom: 2 - 3x + 2y = 0 ji
+x
- 2y
=0
x(0) = a1,
y(0) = a2,
i ( 0 ) = 61,
j(0)=
(4.9.29)
b2.
Taking as the field x = U(t,x , y , z), where y = z, we find that the basic field equation takes the form
au + z -au + ( - x + 2 ~ )au - - 3~
au
- + U-
ax
at
ay
az
+ 2y = 0.
(4.9.30)
By a procedure similar to the method used for solving the two previous problems, the complete solution of this equation is found to be
u=x
cosh t + C1 sinh t + 2Cz cosh 2t + 2C3 sinh 2t sinh t + C1 cosh t + CZsinh 2t + C3 cosh 2t
+
cosh t + C1 sinh t - C2 cosh 2t - C3 sinh 2t 2y sinh t + C1 cosh t + CZsinh 2t + C3 cosh 2t
208
4. Motion of Conservative and Nonconservative Dynamical Systems
sinh t 2C1 cosh t + sinh 2t + C3 cosh 2t '-2sinh t + C1cosh t + sinh 2t + C3 cosh 2t -
+
C2
C2
+ sinh t + C1cosh t + C2 sinh 2t + C3 cosh 2t
*
(4.9.31)
Applying the initial conditions and expressing C4 in terms of C1, CZand C3, we find the conditioned momentum in the form x = p i = Q ~ , y , ~ , a i , a z , b i , bC z ,i , C2, C3)
=x
cosh t + C1 sinh t + 2C2 cosh 2t + 2C3 sinh 2t sinh t + C1 cosh t + C2 sinh 2t + C3 cosh 2t
+
+
+
cosh t + C1 sinh t 2y sinh t + C1 cosh t +
C2 cosh C2
+ C2 sinh 2t + C3 cosh 2t + C3 cosh 2t + 2a2) + cl(bl + 2b2) + C2(-2al + 2a2) + C3(bl - b2) sinh t + C1 cosh t + C2 sinh 2t + C3 cosh 2t
-2 sinh t - 2C1 cosh t sinh t + C1 cosh t + -(a1
2t - C3 sinh 2t sinh 2t + C3 cosh 2t
C2 sinh 2t
(4.9.32)
It is easy to verify that for C1 =
-a=uo
C2
= C3 = 0, the algebraic equations
(i = 1,2, 3 )
aCi
(4.9.33)
are equivalent to the following three equations of motion:
x
+ 2y - (a1 + 2a2) cosh t - (bl + 2b2) sinh t = 0;
x(2 sinh t cosh 2t - cosh t sinh 2t) -
+ cosh t sinh 2t) + 32 sinh t sinh 2t + 2a2) sinh t + (al + 2a2) sinh 2t = 0;
2y(sinh t cosh 2t
+ (-2al
x(2 sinh t sinh 2t - cosh t cosh 2t)
+ cosh t cosh 2t) + 32 sinh t cosh 2t + (bl - b2) sinh t + (a1 + 2a2) sinh 2t = 0. -
2y(sinh t sinh 2t
Finally, the last equation is available from (4.9.32) for C1 = CZ = C3 = 0: x = p1 = x coth t
+ 2y coth t + (al + 2az)/sinh t.
4.10. Applications of the Complete Solutions of the Basic Field Equation
209
4.10. Applications of the Complete Solutions of the Basic Field Equation to Two-Point Boundary-Value Problems
In this section we shall briefly demonstrate that the foregoing theory can be easily adapted to dynamical problems that are formulated as two-point boundary-value problems. For the sake of simplicity, we will confine our attention to problems with one degree of freedom and with differential equations of motion
x. = -aH ,
P
aP
aH
= --
ax
+ Q(t,x,p)
(0 It
IT),
(4.10.1)
together with the boundary conditions (for t = 0), (for t = T).
fi(t,x,p) = 0
f&, x , p ) = 0
(4.10.2)
If we know a complete solution (4.10.3)
P = W t ,x, c1,C2)
of the corresponding basic field equation
au a w a u aH + -+ - - Q(t,x, U)= 0 , at ax au ax
-
(4.10.4)
then by applying (4.10.3) at the initial and terminal times t = 0 and t = T and eliminating the constant CZ, we arrive at an equation of the form W[X(O),p(O), 0, Cil = %[x(T),P ( T ) ,T, Cil,
(4.10.5)
where Y1 and YZ are the solutions for CZwhen (4.10.3) is substituted into (4.10.2). The condition that this relation be satisfied for arbitrary values of C1 will lead to the missing algebraic relation that, together with the equations (4.10.2), form a complete system of algebraic equations for finding x(O), p(O), x ( T ) and p ( T ) . That is to say, the two-point boundary-value problem is transformed into an initial-value problem. This method is illustrated with the following examples. 4.10.1. Stefan-Boltzmann Thermal Radiation Problem The steady-state Stefan-Boltzmann thermal radiation problem is defined as a two-point boundary-value problem (see [17]):
x-x=o x(o) = 1,
(OSt$l), i(i)
+ x4(1)
=
0.
(4.10.6)
210
4. Motion of Conservative and Nonconservative Dynamical Systems
Taking as a field x = p = U(t, x), the basic field equation is of the form
au
au
-++--x=o. at ax
(4.10.7)
The complete solution of (4.10.7) is
+ Cl) + sinh(tc2+ Cl)
(4.10.8)
+ Cl) - xcosh(t + Cl).
(4.10.9)
p = xcoth(t
or C2 = p sinh(t
For the initial and terminal values of t, we find that (4.10.5) becomes p(0) sinh CI - x(0) cosh C1 = p(1) sinh(1 + Cl) - x(1) cosh(1 + Cl). (4.10.10)
Since this relation should be valid for arbitrary values of C1,it follows that p(0) = p ( 1) cosh 1 - x( 1) sinh 1,
x(0) = -p(l) sinh 1 + x(1) cosh 1
(4.10.1 1)
by using well-known properties of the hyperbolic sine and cosine. These two relations together with the given boundary conditions x(0) = 1,
p(i)
+ x4(1) = o
(4.10.12)
form a closed system for obtaining the missing initial values x(O), p(0) or x(l), p(1). Combining (4.10.12) with the second equation of the system (4.10.11), we find that 1 - 0. (4.10.13) x4(l) + x(1) coth 1 - -sinh 1 A root of this equation is found to be x(1) = a - 0.5686. Consequently, ~ ( 1 =) p = -a4 = -0.1045. Repeating the same process as previously described, for the initial-value problem x(1) = a,p(1) = p, we find the solution of the problem to be x = 0.5686 cosh(t - 1) - 0.1045 sinh(t - 1). 4.10.2. Diffusion in a Tubular Reactor
Consider the material balance for a tubular reactor with axial diffusion. The mathematical model is formulated as a two-point boundary-value
4.10. Applications of the Complete Solutions of the Basic Field Equation
21 1
problem (see [18], p. 322): x - 2AX - w2x = 0
1 x(0) - -*O) 212
(0 5 t 5 l),
(4.10.14)
X(1) = 0,
= 0,
where x is the dimensionless concentration, t is the axial coordinate, and A and w are given constants. Denoting p = x = U(t, x), the basic field equation is
au
-au + at
u--212u-w2x=o.
(4.10.15)
ax
The complete solution of this equation is given by p =U
=
[A
CZ ext + n coth(nt + Ci)]x + sinh(nt + Cl)’
(4.10.16)
where n = (A2 + w2)’”. Since C2 is a constant, we can calculate it from (4.10.16) and apply the result at t = 0 and t = 1. Hence, C2 =
1
:[
[ b ( O ) - A] sinh C1 - n -p(0) + 1 cosh C1
= -x(l)e-’[(A
cosh n
+ n sinh n) sinh C1
+ (A sinh n + n cosh n) cosh Cl]. Equating terms with sinh C1 and cosh C1, we obtain ip(0)
n
--p(0) 2A
+ e-’(12
cosh n
+ n sinh n)x(l) = 12
+ e-’(Asinhn + ncoshn)x(l) = n.
(4.10.17)
Therefore, by solving this system, we find the missing boundary values : x(1) =
2~ne’
(n2 + A2) sinh n
+ 2An cosh n ’
2A(A2 - n2)sinh n ’(O) = (n2 + A?) sinh n + 2An cosh n
= X(0).
Finally, from the boundary condition x(0) = X(0)/2A, we can find x(0) and the problem is reduced to an initial-value problem.
212
4. Motion of Conservative and Nonconservative Dynamical Systems
4.10.3. An Eigenvalue Problem
Consider the following eigenvalue problem (see [ 191):
2
1 + -kx t4
x(a) = x(b) = 0
= 0,
(a d t 5 b).
(4.10.18)
We again take x = U(t, x), and the basic partial differential equation is
au
au
kx
-+u-+-=o, ax
at
(4.10.19)
t4
whose complete solution is found to be (see [13]):
.
c 2
(4.10.20)
Applying the boundary conditions, we find that
+ U(b, O)b sni)(:
- U(a, O)a sin($)
=
0.
This homogeneous set of linear algebraic equations has a nontrivial solution under the condition
fi
acos-
a
-bcos-
b
4z b sin -
I&
-a sin a
=
0,
b
which gives sin($
4z
-
$)
=
0.
From this we find all possible values for k: (4.10.21)
for n an integer. To find the corresponding eigenfunctions, we first apply one of the given boundary conditions, say x(a) = 0, to Equation (4.10.20).
213
4.11. Potential Method of Arzhanik’h for Nonconservative Dynamical Systems
Calculating CZand inserting it into (4.10.20), we find that
U(a,O)acos[(&/a)
+C~I
It is easy to show that the equation aU/aCl = 0 is equivalent to
U(a,O)a fi(t x=t sin( at a ) )
fi
This expression, along with (4.10.21), produces the eigenfunctions of the problem; they have the form
x,, = Dnt sin
1
1 L
nnb(1 - a / t ) b-a
(Dn= const.).
J
Together with the eigenvalues (4.10.21), these form the solution of the problem. 4.11. The Potential Method of Arzhanik’h for Nonconservative Dynamical Systems
In this section we will present a field method suitable for the study of a nonconservative dynamical system that was introduced by Arzhanik’h [20] and called the potential method by him. Arzhanik’h’s approach is based on the fact that we are always able to convert an arbitrary system of first-order ordinary differential equations, xi = P ( t , x l , ...,x“)
(i = 1,
...,n),
(4.1 1.l)
into Hamilton’s form by introducing the redundant momenta pi and Hamilton’s function H . To do this we use the relation
H = piFi(t,x l , ...,xn),
(4.1 1.2)
where the summation convention with respect to repeated indices is assumed. It is obvious that the first group of Hamilton’s canonical equations, (4.1 1.3) are identical with (4.11.1).* The second group of canonical equations is
pi = -Pj
aFj 3
(i,j = 1,
..., n).
(4.11.4)
‘According to P. Appell[21], this form of Hamilton’s description was introduced by Liouville.
214
4. Motion of Conservative and Nonconservative Dynamical Systems
Without any change in (4.11. l), we can also introduce the Hamiltonian function. It takes the form
N
.. ., x") + f ( t , X I , ..., x"),
= piFi(t,X I ,
where f is an arbitrary gauge function depending on time t and the state coordinates X I , ...,x". We choose the momentum vector pi (i = 1, ...,n) to be a gradient (covariant) vector of a scalar function s = S(t, X I , ...,x"):
as
Pi = 2.
(4.11.5)
Then the Hamilton-Jacobi partial differential equation, (4.11.6)
for the Hamiltonian function given by (4.11.2) is of the form
as + 7 as F'(t, . xl, ...,x") = 0. at ax
(4.11.7)
-
Note that this partial differential equation simply states that the scalar function S = S(t, X I , .. .,x") is a conservation law of the dynamical system whose differential equations of motion are given by (4.11.1). According to Arzhanik'h's method, we transform the partial differential equation (4.11.7) into a new field equation that is obtained in the following way. We introduce the new field function Y ( t ,P I , ... , a n ) by means of the Legendre transformation of the function S(t, X I , ...,x") (as discussed in Section 4.2): S(t, x ' ,
...,x")
= pixi - Y ( t ,P I ,
- -. pn),
(4.11.8)
where p i = as s ,
i
X = -
av api'
as -- --ay at
at
*
(4.11.9)
Therefore, the partial differential equation (4.11.7) becomes (4.1 1 .lo)
This nonlinear partial differential equation plays a central role in Arzhanik'h's theory.
4.11. Potential Method of Arzhanik’h for Nonconservative Dynamical Systems
215
Let us suppose that a complete solution of the partial differential equation (4.11.10) is known, say,
where the Ci are arbitrary constants. A complete solution of (4.11.10) should have n + 1 arbitrary constants C i , however, one is of an additive nature, since only the partial derivatives of Y appear in (4.11.10). We will now demonstrate that a theorem similar to the Jacobi theorem given by (4.3.16) is valid in this case. That is to say, the general solution of the “canonical” differential equations (4.11.1) and (4.11.4) is given by (4.1 1.12) where Bi (i = 1, ..., n) are a new set of arbitrary parameters. As in the classical Hamilton-Jacobi method, we require that the following condition be satisfied:
The proof is simple. Totally differentiating the second relation in (4.11.12) with respect to time, we find that
a2y p , = 0. +i at aCi ap,
a2\y X
(4.1 1.14)
Since the expression (4.11.10) is an identity for a complete solution of the form (4.11.1 l), we can take the partial derivatives of (4.11.10) with respect to the Ci: a2y --
at aci
aFk a2y axs a p , a c i *
- Pk-
(4.11.15)
Combining the last two relations, we have a2y aCi ap,
-(bs
+ pk$)
=0
( i , k , s = 1, ..., n).
(4.11.16)
Interpreting this system as a system of linear algebraic equations for p , pk(aFk/ax3, we conclude that under the condition (4.11.13), the system of differential equations (4.11.4) will be satisfied. Similarly, to show that the dynamical equations (4.11.1) are satisfied, we first find the total
+
216
4. Motion of Conservative and Nonconservative Dynamical Systems
time derivative of the first relation in (4.11.12)and employ (4.11.15): . X I
a2y =api at
+-apia2y ap,
P S
(4.11.17) On the other hand, differentiating the identity (4.11.10) partially with respect to p i , we obtain
(4.11.18) Combining these last two relations, we see that the differential equations (4.11.1)are also satisfied. This completes the proof. Note that in practical applications, the existence of the superfluous variables pi makes the process of finding the general solution of the dynamical system (4.11.1)rather complicated. Namely, from the first relation (4.11.1 l), we find that
In order to find the solution in explicit form as a function of t, we have to eliminate the momenta pi from the second relation in (4.11.12):
And, to obtain a general solution of (4.11.1)of the form
(4.11.21) xi = f ' ( t , a1, .. ., an), a new set of constant parameters 0 1 , ...,an are defined as functions of Ci and Bi:
ai =
ai(C1, ..., Cn, B1, ...,Bn).
(4.11.22)
In order to illustrate the foregoing theory, we discuss the following two examples, which were also considered by Arzhanik'h [20]. 4.11.1. A Nonlinear System
Consider the nonlinear dynamical system where the pi are given constants.
4.11. Potential Method of Arzhanik’h for Nonconservative Dynamical Systems
2 17
The field equation (4.11.10) is of the form
c
-ay =
n p k P k ( E 7 .
at
k=l
(4.1 1.24)
This equation admits a separation of variables:
c n
=
where each
Yk
k=l
yk(f,Pk),
(4.11.25)
(k = 1, ...,n) satisfies the equation (4.11.26)
The summation convention with respect to repeated indices is not applied. Taking the \Yk to be of the form y k
=
Ckt
-k e k ( P k ) ,
we find that Pkpke”
=
c k ,
which means that ek =
2( C k P k / P k ) ‘ I 2 .
Therefore, a complete solution of (4.11.24) is
c n
=
Ckt
k=l
+2
According to (4.1 1.12), we have xk
=ay = apk
In order to eliminate the momenta
n k= 1
(ckpk/Pk)1/2.
(--y’2.. p k ,
(4.1 1.12) :
(4.11.27)
(4.11.28)
we use the second equation in
alu
-= B k , ack
which, in this case, is equivalent to (4.11.29)
218
4. Motion of Conservative and Nonconservative Dynamics1 Systems
Substituting this into (4.11.28),we find that xk =
and, in this case,
ak
1 Pk@k
-
(4.11.30)
t)
= Bk.
4.11.2. A First Order Equation
Consider a single equation of the first order: x = x.
(4.11.31)
The corresponding field equation is
ay ay -=p-, at ap
(4.11.32)
and a complete solution is found to be Y = Ct
+ Clnp.
a P'
B
(4.11.33)
Hence, aY aP
x=-=-
a\y = -=
Therefore, C
x = - -B
e e
-t -
ac
ae',
t
+ lnp. (4.11.34)
where
a = Ce-B. 4.12. Applications of the Field Method to Nonlinear Vibration Problems*
In this section, we apply the field method described in Sections 4.6 through 4.10 to nonlinear vibration problems, combining the method described here with the well-known multiple-scale asymptotic method (see [lo], [22]). For the sake of simplicity, we confine our considerations to autonomous dynamical systems with a single degree of freedom, whose canonical *The material discussed in this and the next section can be found in References [23] through ~51.
4.12. Applications of the Field Method to Nonlinear Vibration Problems
219
differential equations of motion are of the form X
=p,
= -O;X
+ Ee(x,p),
(4.12.1)
where p is the momentum, x is the generalized coordinate, E is a small constant parameter that represents a measure of the relative importance of the nonlinearity, and 6’ is a given function depending on x and p . Since both variables play an equal role, we have two possibilities for choosing the field variables. We shall consider, in some detail, each of the two possible cases. 4.12.1. The Field Momentum Approach
First, we suppose that the momentum p can be represented as a field depending on time t and the generalized coordinate x : (4. 2.2)
p = X = a,(t,x).
For this case, the basic partial differential equation is of the form aa,
-
at
aa, + a+ O ~ X= Ee(x,p). ax
(4. 2.3)
It is assumed that we are not immediately able to find an exact complete solution of this field equation. Therefore, we will try to establish a method for finding an approximate asymptotic solution by taking advantage of the fact that the term on the right-hand side of (4.12.3) is small. Let us assume the following asymptotic expansions for @ and x in terms of the small parameter E :
+ &@,(XI,T , r) + E’@Z(XZ,T , r) + x(t, E ) = xo(T, r) + Ex,(T, r) + E’xz(T, r) + ...
iP(t,x, E ) = OO(XO,T , r)
(4.12.4) (4.12.5)
where T and r are the “fast” and “slow” time scales, defined T = t(l
+ O~E’ + 0
3
+~
* * a ) ,
~
(4.12.6) (4.12.7)
t = Et.
The quantities CUZ, 0 3 , ... are unknown constant parameters. The general procedure to be followed can be considerably simplified if we impose the following requirement on the linear and unique dependence of the field momentum components Oi of X i ( i = 1,2, ...,n): aa, - aa0- aa,, ax ax0 ax,
amn ... = axn
(4.12.8)
220
4. Motion of Conservative and Nonconservative Dynamical Systems
The time derivative is transformed according to
a
- = (1
at
+02E +W E 3 + 2
a**)-
a + E- a aT at'
(4.12.9)
In order to assure the validity of (4.12.1), we find the total time derivative of (4.12.5) and equate it to (4.12.4). By equating like powers of E , we obtain the following compatibility conditions:
axo aT
@O(XO , T, T ) = -,
(4.12.1 0) (4.12.11)
ax2 axo axl @2(x2,T, t ) = -(T,t )+ 0 2 -(T,7) + -(T, TI, aT aT aT
(4.12.12)
and so on. Since, according to (4.12.2), we have nominated the momentum as a field function, it is natural to suppose that the quantities axl/aT, ..., ax,,/aT can be interpreted as new field functions depending on T, t and the corresponding X i . That is to say,
axl = @ ? ( X i , r, t ) , aT (4.12.13)
ax, _ - a,*(~,, T, TI. aT
Therefore, the compatibility conditions (4.12.10) through (4.12.12) become (4.12.14)
axo (92(x2,T, t )= M ( x 2 , T, t)+ 02-(T, aT
t)
axl + -(T, at
t ) , (4.12.16)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
221
From the last two equations, we have, by partial differentiation,
(4.12.17)
But on the basis of (4.12.8), we also have (4.12.18)
Using Equations (4.12.14) through (4.12.16) and (4.12.9), substituting (4.12.4) and (4.12.5) into the basic field equation (4.12.3), and then equating coefficients of like powers of E , we obtain the following chain system of partial differential equations:
ao0+ 0 0 -d o o + 002x0 = 0; aT axo
a@;
aT
(4.12.19)
azxo +--) axo ao0 (=+= aT axo
a@., a@; + 0;- ax2 + 002x2 = -0 ax,
at
-
a
axo
azxl aTat
(4.12.20)
a@? a2x0 a t at2
&(T, t,XO,@o, XI, @?I;
(4.12.21)
where 81, 82, ... denote the expansions of the function 8 into powers of E . The right-hand sides of these systems are known functions of both T and t at each stage of the development. The system of the partial differential equations (4.12.19) through (4.12.21) can be solved in the sense that, for the each equation of the system, we can find a complete solution. After adjusting it to the given initial conditions, we apply the rule (4.7.7), which produces a corresponding correction term for x. Obtaining xi = Xi(T, t)in this way and substituting it into the corresponding complete solution, that is, calculating the momentum @'! along the trajectory, we also obtain the related correction term for the momentum. Putting this into the next equation of the system, we repeat the calculations.
222
4. Motion of Conservative and Nonconservative Dynamical Systems
Because of the enormous increase in the algebraic calculations in each new step, we suppose, as is generally accepted in perturbation theory, that only a couple of the correction terms in the expansions (4.12.4) and (4.12.5) are necessary for a reasonably good approximate solution to the problem. In order to find a complete solution of Equation (4.12.19), we assume that it takes the form @O(XO, T, T ) = xofo(T) + FoV, t ) ,
(4.12.22)
where f o ( T ) and Fo(T, t ) are unknown functions, Inserting this into (4.12.19) and equating the terms containing xo and the free terms to zero, we arrive at the following system: dfo dT
-
+ f i + 00'= 0,
aF0
a T + fOF0
= 0.
(4.12.23)
COS(UOT + C)'
(4.12.24)
Integrating, it follows that f o = -00
tan(o0T + C ) ,
Fo =
where C is a constant and D is a function of the slow time t. Therefore, we have a complete solution of (4.12.19) in the form Po
= @O =
-x000 tan(ooT + C ) +
D(t) COS(COOT + C)
(4.12.25)
Let the initial conditions applied to the system (4.12.1) be 4 0 , E ) = a,
or
P(0, &) = B,
xo(0) = a,
Xi(0) =
PO(O) = P,
Pi(())= 0,
0,
2.26)
2.27)
for i = 1,2,3, .... Since we are interested in the conditioned form of the solution of the basic equation (see Section 4.7.2), we insert (4.12.27) into (4.12.25) and obtain D(0) = p cos C + a 0 0 sin C , (4.12.28) where we have used the obvious relation @0(0,0, a) = p, which follows from the nomination of the field variable p = @ ( t ,x). The form of Equation (4.12.8) indicates the structure of the slow function D(t), which is selected in the spirit of the variation of the arbitrary constants, D(t) = B ( t ) cos C + A ( @sin C . (4.12.29)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
Comparing the last two equations for T B(0) = p,
=
223
0, we find, by initial conditions,
A(0) = a o o .
(4.12.30)
It is to be noted that the guiding principle that determines A(7) and B(r) is that @ I is to be free of secular terms of the type T sin o o T and T cos OOT, because they lead to unboundedness with respect to T. The differential equations thus obtained should be integrated subject to the initial conditions (4.12.31). Using (4.12.29) for D(s), the conditioned form of the momentum (4.12.25) becomes @O(XO, T, 7) =
-xooo tan Y
where
+ B(r)cos Ccos+YA(r)sin C , (4.12.31)
+
Y = COOT C .
(4.12.32)
According to the rule (4.7.6), we can easily verify that the equation
a@o - -0 ac
(4.12.33)
will lead to the equation 00x0
= A(7)cos o o T
+ B(t)sin o o T
(4.12.34)
In fact, taking into account the following identities: cos c COS(CUOT + C ) = cos o o T + sin o o T tan(o0T + C ) sin C COS(COOT + C ) = -sin o o T + cos ooT tan(ooT + C ) , we can write (4.12.31) as
40(x0, T, r) = tan(w0T + C)[-XOOO + A(7)cos COOT+ B(t)sin COOT]
+ B(T)cos OOT,
- A ( t ) sin COOT
and the equation (4.12.34) is immediately verified. At the same time, inserting (4.12.34) into the last equation [or in (4.12.31)], we find the momentum along the trajectory: po(T, r) = 4oIx0= - A ( t ) sin o o T + B(T)cos COOT. (4.12.35)
224
4. Motion of Conservative and Nonconservative Dynamical Systems
Substituting (4.12.31) and (4.12.34) into the right-hand side of (4.12.20), we obtain the basic field equation in the form
a+?
+ 4?+ oixl aT ax1
=K
(
dA dB A , B, - - T, 7 , C ) d7’ d7’
(4.12.36)
where K(A, B, d A / d t , dB/dr, T, 7 , C ) denotes the group of terms (4.12.37)
In order to find a complete solution of (4.12.36), we put
4:
= fo(T)xi
+ K ( T ,7),
(4.12.38)
where, according to the requirement (4.12.18), the function fo(T) is given by the first relation in (4.12.24) and Fl(T, 7 ) is an unknown function of both T and 7 . By substituting (4.12.38) into (4.12.36), we obtain
am
-=
aT
dA dB K ( A , B, -, -, T, t, d7 d7
+gcosooT+
(4.12.39)
where (4.12.40)
As mentioned previously, the functions A(7) and B(7) may be obtained by the requirement that no secular terms appear in the differential equations. Integrating (4.12.39) and repeating the same process as in the previous step, we find @ ? ( X I , T, t, C )and match it with the initial conditions. Applying the rule aO?/aC = 0, we obtain x1= xl(T, 7 ) . Next we calculate the momentum and substitute these quantities into the rightalong the trajectory, @? hand side of (4.12.21). The same process of calculations is then repeated.
Ix1,
THEVAN DER POLOSCILLATOR As an illustration of the previous theory, we turn to van der Pol’s nonlinear and nonconservative oscillator : x =p,
p
=
-x
+ E(1 - x Z )p .
(4.12.41)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
225
Taking the field momentum approach, we put p = @(t,x), the basic partial differential equation is then
a@
a@ + x
- + 0at ax
= E(1 - x2)@(t,x).
In this case, we have 00
0 = (1 - X2)@(t,x),
= 1,
Y =T
(4.12.42)
+ C;
(4.12.43)
therefore, using (4.12.3) and (4.12.4), we find that
el = (1 -
(4.12.44) (4.12.45)
From (4.12.31), we have
+ dA/dtsin C cos Y
a@,- - dB/dtcos C at
dB dA =-cosT--sinT+ dr dr
tanY
.
(4.12.46)
In evaluating this expression, we have employed the identities given after Equation (4.12.34). The quantity K defined by Equation (4.12.37) now becomes
dA dr
= 2-sin
T-
- [l - (A cos T + B sin T)’]( -A sin T + B cos T),
(4.12.47)
and Equation (4.12.38) becomes dA dt
aMl = k-sin aT
-
dB T - 2-cos dr
T
I
[I - (A cos T -k B sin T ) 2 ] ( - Asin T + B cos T) cos Y
[ (2
A3 1 = sinT 2--A+-+-AB2 4 4
1
+ 4-1 cos 3T(B3 - 3d2B)
- 3AB2) cos Y .
(4.12.48)
226
4. Motion of Conservative and Nonconservative Dynamical Systems
Had we integrated this equation with respect to T, considering the functions A ( t ) and B(7) as constants, the terms containing sin T c o s Y and cos T cos Y (with Y = T + C ) would have produced secular terms of the form T sin T and T cos T. In order to avoid this, we choose A 2 + B 2) = O ,
2 dA --A(ldt
A 2 + B 2) = O .
- 2 -dB +B(ldt
(4.12.49) Multiplying the first equation by A and the second by B and subtracting, we find that
);
d dt
-(d)-
-
=
+ B2(t).
0,
a2 = A 2 ( t )
(4.12.50)
Integrating this equation, it follows that a2 is of the form 4
2
a =
1
+ [(4/a02) - 1 1 e - ~ ’
a? = A(0)2+ B(0)2. (4.12.51)
Now we introduce the so-called van der Pol variables, A ( t ) = a(t)cos a,(?) and B ( t ) = a(r) sin a,(t), where a(t) is the amplitude and p(t) denotes the phase angle. Substituting these into (4.12.49), we obtain da,/dt = 0, or a, = a , ~= const. Therefore, Equations (4.12.31), (4.12.34) and (4.12.35) become, respectively,
00= -xotanY
C* + a-sin COSY ’
(4.12.52)
xo =acos T * , PO
=%Ixo
(4.12.53)
= -asin T * ,
(4.12.54)
with C* = C - 90,
T*
=
T+
~ 0 ,
Y = T + C = T*
+ C*.
(4.12.55)
Equation (4.12.48) becomes aMl - -
aT
*a3 sin 3T* cos Y = *a3 sin 3T* cos(T*
+ C*).
(4.12.56)
Integrating with respect to T * , taking into account that dT = dT* and
227
4.12. Applications of the Field Method to Nonlinear Vibration Problems
keeping in mind that 0;= fox1 @?(XI,
T, r) =
-XI
+ (Ml/cos Y ) , we find that
1 tan Y - - a 16
cos(2T* - C*) + cos(4T* cos Y
+ C*) (4.12.57)
Observing (4.12.53), we can write (4.12.15) in the form Ql(x1,
da
T, 5) = @?(.XI,T, r) + -cos dr
T*.
(4.12.58)
Hence, since xl(0) = Q1(O) = 0, we find da Qf(0) = - -(0) cos @I, dt
and from (4.12.57) for t = 0 (i.e., T = 0, D~(o)=
da -z (0)cos
90
cos C
T =
0), we have
+ & ~ ( o ) ~ [ c o -s (C~ )~+ 9 COS(3po + c)]. (4.12.59)
As in the previous step, we assume that the function DI(T)is equal to D I ( T )= BI(T)cos C*
+ AI(T)sin C*,
(4.12.60)
where Bl(r) and AI(T)are unknown functions of the slow time, to be determined in the next step of the calculation. Putting r = 0 in (4.12.60) and then comparing the terms containing sin C and cos C with those in Equation (4.12.59), we can easily find Al(0) and Bl(0). Combining (4.12.60) and (4.12.57), we have 1 Ql(x1, T, T ) = -XI tan Y - - a 16
C O S ( ~ T-* C*) + ~ c o s ( ~ T+ *C*) cos Y
) C* + B1(r) cos C*cos+YA I ( Tsin
(4.12.61)
It is easy to verify that the equation aQl/aC* = 0 is equivalent to XI
= AI(T)cos T*
a3 + Bl(r) sin T * - -sin 32
3T*.
(4.12.62)
228
4. Motion of Conservative and Nonconservative Dynamical Systems
Substituting (4.12.62) into (4.12.61), we find the first correction term for the momentum along the trajectory:
@?Ix,
= Bi(t) cos T* - Al(7) sin T* - &sin 3T*.
(4.12.63)
Note that in calculating (4.12.62) and (4.12.63), the identities given after Equation (4.12.34) can be of considerable value. To find a complete solution of the basic field equation (4.12.21), we assume that the solution is of the form @2
= fo(T)x2
+ F2(T, t) = -x2
+ M2(T’ ‘I. cos Y
tan Y
(4.12.64)
The partial differential equation (4.12.21) now becomes
a~~
1 -COSY a T
aTat
ar
at2
(4.12.65)
Substituting (4.12.52), (4.12.53), (4.12.57) and (4.12.63) into this equation, it follows that *=[(2ao,-2--aT
dB1 dt
- - a 3 ,da 1 a 2B l + B l + - da 4 dt 4 dt
3 5 + -a2A1sin3T* + -a5cos5T* 4 128
I
COSY.
-
$a2)
(4.12.66)
Elimination of the secular terms necessitates satisfying the following equations: d2a dBi 2- Bl(1 - *a’) = 2ao2 - 7(1 dt dt
dA 1 2- Al(1 - $a2) = 0. dt
+ &a5;
(4.12.67) (4.12.68)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
229
By means of (4.12.50), we can express the last two equations as follows:
dB1 2 da 2--.---B d7 a d7
1
-
dA 1 d7
2a(wz
(:$
da + &-) - (&a2 - $)--. d7'
- l>,,
= 0.
(4.12.69) (4.12.70)
In order to avoid the appearance of secular terms, we choose 0 2
(4.12.71)
=16 1.
Now, integrating (4.12.69) and (4.12.70), we find
+ + a h a - &a3,
Bl(7) = -bla
Al(7) = ala3e-',
(4.12.72)
where a1 and bl are constants of integration. The first correction terms, (4.12.62) and (4.12.63), now become XI =
a1a3e-'cos T*
O I =~-&cos3T* ~ ~
+ ( i a l n a - bla - &a3) sin T* - &a3sin 3T*;
(4.12.73)
+ ( i a l n a - bla - &a3)cos T* - ala3e-'sin
T*. (4.12.74)
Therefore, for a second approximation,
x = acos T* + &[a1a3e-'cos T* - &sin3T*] + 0 ( e 2 ) ,
= -asinT* -
+E
[
&a3 sin 3T*
-&cos3T*
da + -cos d7
T*=t(1-;),
7=&t,
(4.12.75)
+ ( i a l n a - bla - &a3)sinT*
1+
T*
where
+ ( i a l n a - bla - &a3)cos T*
0(c2),
a=
(1 +
(4.12.76)
2 [(4/ao2) - l ~ e - ' ) " ~ ' (4.12.77)
This approximate solution is in full agreement with the results obtained in Nayfeh's monograph [l 11 by means of the method of multiple scales.
230
4. Motion of Conservative and Nonconservative Dyoamical Systems
4.12.2. The Field Coordinate Approach
As a second possibility, consider the case in which the coordinate x plays the role of the field variable; x = U(t,p). (4.12.78)
The basic field equation corresponding to the dynamical system (4.12.1) now takes the form
au + au [O;U+ EB(U,p)] - p
-
at
ap
(4.12.79)
= 0.
Let us assume the following asymptotic expansions for small values of the parameter E : U(t,p,E )
=
Uo(po, T, T) + EUi(p1, T, 7) + E ~ U ~ (T,PE )~+,
p(t, E ) = po(T, 7 )
+ EPI(T,7 ) + c2p2(T,4 + .-.,
*.*,
(4.12.80) (4.12.81)
where T and 7 the “fast” and “slow” times defined by (4.12.6) and (4.12.7). As with the other field variable, we simplify the analysis by assuming that (4.12.82)
Substituting (4.12.80) through (4.12.81) into (4.12.79) and equating like powers of E , we obtain the following system of basic field equations:
avo
aT
auo
(4.12.83)
OoUo-Po = 0; apo
auo- 91 -, avo. aT at aP0 au2 woU2-au2- p2 = -02- auo- aul - 92-; auo --
aul
--
aT
aul
OoU1-p1 = aPl
ap2
--
aT
a7
apo
(4.12.84) (4.12.85)
where 0 1 , 1 9 2 , ... denote the terms of the expansion of the function 9. It is clear that this system of partial differential equations has a considerably simpler structure than the system (4.12.19) through (4.12.21), due to the absence of the compatibility conditions (4.12.14) through (4.12.17). The general plan for obtaining an approximate solution remains unchanged, and it is described by the field momentum approach. For example, the
4.12. Applications of the Field Method to Nonlinear Vibration Problems
231
complete solution of the basic field equation (4.12.83) is found to be Po Uo(p0, T, t) = -tan 0 0
Y
where C is a constant and Y = woT B ( t ) satisfy the initial conditions
+ A ( @cos Ccos-YB(t)sin C ,
(4.12.86)
+ C . The unknown functions A ( t ) and Po = -. P B(0) = -
A(0) = a = xo(O),
(4.12.87)
wo
0 0
In order to illustrate this technique, we consider the following example. OSCILLATOR WITHSMALLCUBICDAMPING
Let us consider an oscillator with small cubic damping, whose differential equation of motion and initial conditions are given by References [lo] and [25]: x =p,
x(0) = 1 ,
P = - x - &P3
9
(4.12.88)
P(0) = 0.
With the field variable x = U(t,p),the basic field equation is
au - - u-au - p at
aP
au
= cp3--.
(4.12.89)
aP
To lowest-order terms, the system of partial differential equations (4.12.83) through (4.12.85) for this case becomes (4.12.90)
a ul
--
aT
a
-u2 -
aT
u1-a ul - p 1 = aPl
au2- p2 = -
u2-
aP2
avo+ po-; ,auo at apo
(4.12.91)
--
+ 3p;pi-,avo apo
(4.12.92)
where the notation (aUo/aT)I,, means that the quantity aUo/aT should be evaluated along the trajectory PO = po(T, t),which is obtained from the previous step. The conditioned-form solution is given by (4.12.86) for 00 = 1 . From the same equation, applying the rule aUo/aC = 0 and calculating the field
232
4. Motion of Conservative and Nonconservative Dynamical Systems
variable UOalong the trajectory, we find that
PO = - A ( t ) sin T + B ( t ) cos T,
(4.12.93)
xo = UoL, = A ( r ) cos T
(4.12.94)
+ B(r) sin T,
with A(0) = 1 ,
B(0) = 0 .
(4.12.95)
Substituting (4.12.86)and (4.12.95)into the right-hand side of (4.12.91) and looking for a complete solution of the form UI = PI tan Y
7) + Ml(T, COSY
(Y = T
+ C),
(4.12.96)
Equation (4.12.91) is reduced to aM1 = -dA cos C
aT
dt
=
dB + -sin C + [ - A sin T + Bcos TI3sin Y dr
(2
cos C --
- A 3sin4T + 3A2B sin3Tcos T - 3AB2 sin T cos3T
)
+ B3 sin T cos3T + sin C
(2
- - A 3 sin3Tcos
T
+ 3A2Bsin2Tcos2T - 3ABZsin T cos3T + B3 Integrating this equation with respect to T and considering the slow time as a constant, we find that Ml(T,x) = c o s C -
1
T-A3(-asin2T+&sin4T)
dB
sin C [ [ dt
+ i3B ( A 2
+ Bz)
t
1
(4.12.97)
where Kl(r) is an arbitrary function of the slow time (or U I )is unbounded as T + 00 unless dA dt
-
+ -38A ( A 2 + B 2 ) = 0 ,
dB 3 - + -B(A2 dr 8
+ Bz) = 0 .
t. Note
that Ml(T, t)
(4.12.98)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
233
The solution of this system, subject to the initial conditions in (4.12.95), is A(?) =
2
(4
B(7) = 0.
+ 37)”2’
(4.12.99)
Thus, the expression U I ,given by (4.12.99, becomes VI(PI, T, 7) = P I tan Y - A 3
COS
c(&sin 4 T - 3sin 2T) cos Y
4T + A 3 sin C(&coscos Y
cos 2T)
K(7) +-cos Y’
(4.12.100)
where -&A3(?) sin C + Kl(7) = K(7). Since U1(0,0,0) = x1(0,0)= 0, andpl(O, 0) = 0, we find from (4.12.100) that for t = 0 (that is, T = 0, 7 = 0), we have K(0) = & sin C . As in the previous step, we suppose that the function K(r) can be expressed K(7) = AI(T)cos C - Bl(7) sin C,
(4.12.101)
with Ai(0) = 0,
Bi(0)
=
6,
(4.12.102)
where the slow variable functions A l ( r )and &(?) will be determined in the next step of the approximation. Thus, the conditioned-form solution of (4.12.91) is Ul
=pltanY -
cos C(& sin 4 T - $ sin 2T) cos Y
4 T - # cos 2T) Al(7) cos C - Bl(7) sin C + A3 sin C(& coscos + Y cos Y
(4.12.103)
It is easy to verify that the condition dUl/dC = 0 yields pl(T
7) =
&A3 cos 3 T - &A3 cos T - A1 sin T
+ B1 cos T.
(4.12.104)
Substituting this into (4.12.102), we find that xl(T,
7) =
U1 I p l
= &A3 sin 3 T
+ &A3 sin T + A1 cos T + BI sin T.
(4.12.105)
234
4. Motion of Conservative and Nonconservative Dynamical Systems
Now, from (4.12.93), (4.12.94) and (4.12.86), we have po(T, r) = -A(7) sin T, XO(T, T) =A(?)cos T,
(4.12.106)
A ( t )cos C UO(PO,T, 5 , C ) =POtan Y + cos Y
(Y = T
+ C).
Forming the partial derivative with respect to T in the third equation, having substituting in the result from the first, we obtain A(T)sin C
(4.12.107)
Looking for a complete solution of (4.12.92) of the form U2 = p ~ t a n Y+ cos Y M2(T,
T)
(4.12.108)
and substituting (4.12.103) through (4.12.107) into the right-hand side of (4.12.92), we have, by collecting only the resonant terms,
rz
27 + sinC - + -38 A ~ B--A' ~ 256 + (nonresonant terms).
-
(4.12.109)
To eliminate the secular terms, we require that dA1 dr
-
dB1 dr
-
3 27 + -A2B1 - -A2 8 256
+ -98 A -
~ A= ~ 0,
0 2 A = 0.
(4.12.110)
The solutions of these equations, subject to the initial conditions (4.12.102), are A I ( T )= 0, Bl(t) = -&A3 = -Q(4
Therefore, for t
+ S A + A02t + 3?)-3/2 + $4 + 3r)-1'2 + 202t(4 + 3t)-'l2.
(4.12.111)
-, 00, the function Bl(t) will be unbounded unless 0 2 = 0.
4.13. A Linear Oscillator with Slowly Varying Frequency
235
Thus, we have the following approximations to the second order in P
= Po
+ EPl
= - A ( t ) sin T
E:
+ &[&A3cos 3T + (-%A3 + S A ) cos TI
x = UOL, + U1Ll = xo
+ EX1
+ &[&13sin3T+ (&A3 + %A) cos TI, (4.12.112) where A(r) is given by (4.12.98) and T = t + O(e3).This result is in full = A cos T
agreement with the expansion obtained by Kevorkian and Cole (see [lo], p. 123). Concluding this section, the following two remarks are of interest. First, the asymptotic method demonstrated in this section is based on the supposition, generally accepted in all asymptotic methods, that the successive approximations are developed formally by a recursive procedure for a fixed number of terms and for E -, 0. In other words, we require that the asymptotic expansions (4.12.4) and (4.12.5) represent a reasonably accurate solution for a long interval of time, t + 00. Second, we can easily extend our considerations to nonautonomous systems. Since this extension is straightforward, we will demonstrate it by means of an example in Section 4.13. 4.13. A Linear Oscillator with Slowly Varying Frequency
As an example of a nonconservative, nonautonomous, linear dynamical system, we consider an oscillator with a slowly varying restoring force, whose differential equations are given by x
= y,
(4.13.1)
j , = - K( t) x ,
where t = Et is the slow time and E is a small parameter. We suppose that K ( t ) = 0(1) (in E ) and is strictly positive on the entire time interval. Taking as the field variable x = U(t,y ) , the basic field equation becomes
au - K(7)U-au - y = 0. _ at aY
(4.13.2)
We assume that there exists an asymptotic representation for U and y of the form
U(t,Y , E ) = uo(yo, T, t) + EUICYI, T,
~ ( tE ), = yo(T, 4 + EYI(T, +
*.*
,
+
,
(4.13.3)
236
4. Motion of Conservative and Noneonservative Dynamieal Systems
where we introduce, as suggested in Reference [ l l ] (p. 282), the fast time scale 1 T = -f(r),
(4.13.4)
&
where f(r) is an adjustable function of the slow time 5. Assuming, as before, that aU/ay = aUo/ayo = aUl/ayl = - - -we , obtain the following system of partial differential equations by equating the coefficients of the lowest powers of E to zero: (4.13.5) (4.13.6)
To find a complete solution of (4.13.5), we assume a form similar to that of the previous cases:
UO= A(7)yOtan Y
+M(r) cos Y
(Y = T + C ) ,
(4.13.7)
where A(r) and M(r) are to be determined. Substituting (4.13.7) into (4.13.5), we find that - 1
[
:- 1
+ tanZY A(?) - - K(r)A2(r)
+
(4.13.8)
This relation will be satisfied identically for all Y, yo and M(r) if
df
A(7)-
dr
- 1 =
0.
A(r)-df - K(r)A2(t)= 0, dt df-
dr
From this system, we find that
A(r)K(r)= 0.
237
4.13. A Linear Oscillator with Slowly Varying Frequency
Therefore, (4.13.9)
As discussed previously, we select the function M(r) to be
M(r) = A(r) cos C
-
B(r)sin C .
(4.13.10)
Consequently, the conditioned-form solution of the basic field equation (4.13.5) is UO =
&tan
Y
+ A ( r )cos Ccos-YB(r)sin C .
We easily verify that the condition YO
=
(4.13.11)
auo/ac= o leads to
-m[A(r)sin T - B(r)cos TI.
(4.13.12)
Inserting this into (4.13.1l), we also have xo = UoL,, = A(r)cos T
+ B(r)sin T.
(4.13.13)
Taking into account the requirement aUo/ayo = aUl/ayl, we seek a complete solution of (4.13.6) of the form (4.13.14)
Substituting (4.13.14) and (4.13.11) into (4.13.6) and collecting the secular terms, we find that
( r ) 1 dK --dA - Adr
dB B(r) 1 dK -+--dr 4 K(r) dr
4 K(r) dr
+ (nonsecular terms).
1
Thus, we have
dA dr
1 1 dK 4 K(r) dr
- + - --A(r)
= 0,
dB dr
1 1 dK 4 K(r) dr
- + - --B(r)
= 0.
(4.13.15)
Integrating, we find that (4.13.16)
where D1 and DZare constants of integration.
238
4. Motion of Conservative and Nonconservative Dynamical Systems
Hence, (4.13.12) and (4.13.13) lead to
(4.13.17)
The same result was obtained in Nayfeh’s monograph (see [ l l ] p. 282) by means of a generalized version of the multiple scale method.
References 1. Courant, R. (1962). Methods of Mathematical Physics, Vol. II, Partial Differential Equa-
tions. Interscience Publishers, New York. 2. Amol’d, V. I. (1980). Mathematical Methods of Classical Mechanics. Springer, New York. 3. Ray, J. R. (1973). “Modified Hamilton’s Principle,” American Journal of Physics 41, 1188-1 190. 4. Gelfand, I. M., and Fomin, S. V. (1963). Calculus of Variations. Prentice-Hall, Englewood Cliffs, N.J. 5. DjukiC, Dj. (1975). “A Note on Hamilton’s Principle,” Mathematica Japonicae 20(2), 132-140. 6. Bryson, A. E., Jr., and Ho, Yu-Chi. (1975). Applied Optimal Control. Hemisphere Publishing Corporation, Washington. 7. Meyer, G. H. (1973). Initial Value Methods of Boundary Value Problems. Academic Press, New York. 8. Denman, H. H., and Buch, L. H. (1973). “Solution of the Hamilton-Jacobi Equation for Certain Dissipative Classical Mechanical Systems,” J. Math. Phys. 14(3), 326-329. 9. Vujanovic, B., and Strauss, A. M. (1981). “Linear and Quadratic First Integrals of a Forced Linearly Damped Oscillator with a Single Degree of Freedom,” J. Acoust. SOC. Am. 69(4), 1213-1214. 10. Kevorkian, J., and Cole, J. D. (1980). Perturbation Methods in Applied Mathematics. Springer-Verlag, New York. 1 1 . Nayfeh, A. H. (1973). Perturbation Methods. Wiley, New York. 12. Vujanovic, B. (1981). “On the Integration of the Nonconservative Hamilton’s Dynamical Equations,” Int. J. Eng. Sci. 19(12), 1739-1747. 13. Sutela, T., and Vujanovic, B. (1982). “Motion of a Nonconservative Dynamical System Via a Complete Integral of a Partial Differential Equation,” Tensor, N.S. 38, 303-310. 14. Vujanovic, B. (1982). “Conservation Laws and a Hamilton-Jacobi-Like Method in Nonconservative Mechanics,” In Avez, A., Blaquiere, A., and Marzollo, A. Dynarnical Systems and Microphysics-Geometry and Mechanics. Academic Press, New York. 15. Ince, E. L. (1963). Integration of Ordinary Differential Equations, seventh edition. Oliver and Boyd, Edinburgh. 16. Garabedian, P. R. (1964). Partial Differential Equations. Wiley, New York.
References
239
17. Dyer, P. (1970). The Computational Theory of Optimal Control. Academic Press, New York. 18. Finlayson, B. A. (1972). The Method of Weighted Residuals and Variational Principles. Academic Press, New York. 19. Chuev, M. A. (1975). “The Hamilton-Jacobi Method and a Problem in Solving Ordinary Differential Equations,” Differential Equations. XI(12), 2183-21 88. 20. Arzhanik’h, I. S. (1965). Field of Impulses. “Nauka” Tashkent. 21. Appell, P. (1953). Trait6 de MPcanique Rationelle, Tome Deuxihe. Gauthier-Villars, Paris. 22. Cole, J. D. (1968). Perturbation Methods in Applied Mathematics. Blaisdel, Waltham, Mass. 23. Vujanovic, B. (1984). “A Field Method and Its Application to the Theory of Vibrations,” Int. J. Non-Linear Mech. 19(4), 383-396. 24. Vujanovic, B., and Strauss, A. M. (1984). “Study of the Motion and Conservation Laws of Nonconservative Dynamical Systems Via a Field Method,” Hadronic J. 7 , 163-185. 25. Vujanovic, B., and Strauss, A. M. (1987). “Applications of a Field Method to the Theory of Vibrations,” J. of Sound and Vibration 114(2), 375-387.
A Study of the Motion of Conservative and Nonconservative Dynamical Systems by Means of Field Theory
4.1. Introduction
In this chapter, we consider the problem of integration of the differential equations of motion of conservative and nonconservative dynamical systems by means of certain field theories. In the first part of the chapter, we discuss the Hamilton-Jacobi field method for finding the motion of those dynamical systems that are completely describable by means of a Lagrangian (or Hamiltonian) function. The general idea of this celebrated method is that the problem of integration of the differential equations of motion, which are expressed as a system of ordinary differential equations of the first order (Hamilton’s canonical equations), can be accomplished by finding a complete solution of a single nonlinear partial differential equation of the first order, namely, the Hamilton-Jacobi partial differential equation. More precisely, if we succeed in finding a complete solution of the Hamilton-Jacobi partial differential equation, the motion of the dynamical system can be easily found directly from this complete solution without any additional integration, by using only elementary algebraic operations. Note that the field function figuring in the Hamilton-Jacobi theory is Hamilton’s action, and the differential equations of motion are the characteristics of the HamiltonJacobi partial equation. Naturally, the applicability of the Hamilton-Jacobi method is based upon the premise that it is easier to find a complete solution of the HamiltonJacobi partial differential equation than it is to integrate the ordinary set of the differential equations of motion. Contrary to the widespread opinion that the problem of integrating a partial differential equation is usually more complicated than that of integrating a system of ordinary differential 152
4.2. Hamilton’s Canonical Equations of Motion
153
equations, numerous authors have made it a point to emphasize that the ordinary differential equations of motion “may be difficult to integrate by elementary methods, while the corresponding partial differential equation is manageable” [I]. Recently, Arnol’d [2] made the statement that the Hamilton-Jacobi method is “the most powerful method known for exact integration, and many problems, which were solved by Jacobi, cannot be solved by other methods.” In the second part of this chapter, we make an attempt to establish a field method for studying nonconservative holonomic dynamical systems that do not possess Hamilton’s action integral, and the Hamilton-Jacobi field equation cannot be employed. The basic supposition in this field method is that one of the state variables (a generalized coordinate or a generalized momentum) figuring in the dynamical system can be interpreted as a field depending on time and the rest of the state variables of the dynamical system. The resulting field equation, which we call the basic equation, is a single quasi-linear partial differential equation of the first order. As in the case of the Hamilton-Jacobi theory, the differential equations of motion of the dynamical system are differential equations of the characteristics of the basic field equation. Besides, we will demonstrate that there exists a rather completeformal similarity between the field method presented here and the Hamilton-Jacobi method. Namely, if a complete solution of the basic partial differential equation is available, the motion of the dynamical system can be obtained without any additional integration. Note, finally, that our field method can be applied to conservative dynamical systems (for which the Hamilton-Jacobi method can be also applied) but in this case, the field method is not identical to the Hamilton-Jacobi method. 4.2. Hamilton’s Canonical Equations of Motion Consider a holonomic dynamical system whose Lagrangian function is of the form (4.2.1) L = L(t,x , x), where t denotes time, x stands for the generalized independent coordinates x = (xl, x ” ) , and x = [x’, ..., x”) is the generalized velocity vector.
...,
4.2.1. The Legendre Transformation of L Our first objective is to find a dual (conjugate) function to the Lagrangian function L , which is obtained by means of the Legendre transformation of L with respect to the generalized velocities x i (called the active variables),
154
4. Motion of Conservative and Nonconservative Dynamical Systems
while the generalized coordinates x i and t play a passive role in this transformation. The basic idea of the Lagendre transformation is to find the dual function H(t, x , p ) , named Hamilton's function or the Hamiltonian, in which the partial derivatives of the old (active variables) of the original function L are used as the new independent coordinates p i , namely,
pi =
aL
(i = 1,
..., n).
(4.2.2)
One of the essential points in accomplishing this goal is the requirement to solve (4.2.2) with respect to the old variables .tiand to find n relations of the form
xi = f i(t,x , p ) .
(4.2.3)
Naturally, this inversion is possible under the condition that the following functional determinant (Jacobian) is different from zero, i.e., det()
= det(*)
axi a?
H(t, X , p ) = [pixi - ~ ( Xt ,,XI]
# 0.
i i
=fi(t,x,p)
(4.2.4)
(4.2.6)
(4.2.7) Using (4.2.2) and (4.2.3), the last two terms on the right-hand side of this relation cancel, and we have (4.2.8) These equations are the first group of Hamilton's canonical equations of motion.
4.2. Hamilton's Canonical Equations of Motion
155
It is easy to show, by repeating calculations similar to those used for (4.2.7), that the following relations also follow from (4.2.6): (4.2.9) (4.2.10) It is now straightforward to show that the Legendre transformation is reversible. Namely, if we follow the same procedure as just described, we can demonstrate that for a given Hamiltonian function H = H(t, x , p ) , considering the pi as the active variables, the Legendre's transformation of H = H(t,x, p ) leads to the Lagrangian function for the system, L = L(t, x , x). 4.2.2. The Second Group of Equations
To derive the second group of differential equations of motion, we can proceed in several ways, by using some of the dynamical considerations. For example, we can start from the central Lagrangian equation (which is a modification of D'Alembert's variational principle) discussed in Section 3.10 of Chapter Three. If the dynamical system is subject to nonconservative forces Q;= Qi(t,x , x), then we have, according to (3.10.21), d dt
- @i ax') = 6L
+ Qi 6x' + pi
Note [as we discussed after Equation (3.10.21)] that we are not obliged to suppose that the operators of variation, 6, and differentiation, d / d t , are commutative. Substituting from (4.2.8) into Q; , we express the generalized nonconservative forces in terms of t , x , and p as Q; = Q i ( t , x , p ) . By putting L = pix' - H(t, x , p ) into (4.2.1 l), we obtain
d - ( p i 6 x i ) = 6[piXi - H ( t , x , p ) ] + Q i ( t , x , p ) 6 x i+ pi[(6Xi)*- 6%']. dt (4.2.12) That is to say,
+ p i ( 6 ~ ' ) -p;&' ' + Q;6xi,
(4.2.13)
156
4. Motion of Conservative and Nonconservative Dynamical Systems
or
The group of terms with 6pi vanish because of (4.2.8), and since the 6xi are arbitrary and independent variations, we arrive at the second group of canonical equations: (4.2.15)
The second possibility for deriving these differential equations is to start from Lagrange’s differential equations: (4.2.16)
Using (4.2.2) and (4.2.9) and transforming Qi into Qi, we immediately obtain (4.2.15).
4.2.3. Some Properties of Canonical Equations The canonical, or Hamilton’s canonical, equations of motion, (4.2.17)
form a system of 2n ordinary differential equations of the first order with respect to x i and pi. It is easy to show that if the dynamical system is scleronomic (aH/at = 0) and conservative (Qi = 0), then the Hamiltonian H of the system is a constant of motion. Indeed, differentiating H = H(t,x, p ) with respect to time gives us I? = (aH/axi)xi+ (aH/api)pi, and employing (4.2.17) (with Qi = 0), we have I? = 0, or H(t,x , p ) = const.
(4.2.18)
Note that, according to Equations (4.2.5) and ( 3 . 7 4 , this conservation law is identical with the Jacobi integral discussed in Section 3.7-A of the previous chapter. Also, if Hamilton’s function and the generalized forces Qi do not depend on a particular generalized coordinate, say x s (s is a fixed integer), it follows from the second group of canonical equations in (4.2.17) that p s = const.,
which is the cyclic conservation law discussed in Section 3.7.2.
(4.2.19)
4.2. Hamilton's Canonical Equations of Motion
157
Now we discuss the question of derivation of the canonical equations of
motion,
(4.2.20)
from Hamilton's variational principle. It is well known that, if we wish to derive the Lagrangian equations of motion, we can use Hamilton's action in the form
Z=
1
t1
L(t,x,x)dt,
(4.2.21)
to
and Hamilton's principle asserts that Z is stationary for the actual path of motion. This implies that the first variation of the action integral along the real path vanishes, 61 = 0, which leads to Lagrange's equation, (4.2.16), with Qi = 0. Using relation (4.2.5), we can express L ( t , x , x) in terms of the canonical variables, and Hamilton's action reads
Z=
S
tl
to
[Pix' - H(t, x , P ) ] dt.
(4.2.22)
The variation of this integral demands some care. In Hamiltonian dynamics, the generalized coordinates x i and the generalized momenta pi play an equal role. If we wish to vary the action, (4.2.22), we have to take into account that the variables x i and pi are not independent, but that they have to satisfy the (nonholonomic) constraints (4.2.23)
which arise from the definitions of H a n d p , Equations (4.2.5) and (4.2.2). As proposed by J. R. Ray (see [3]), we can vary x i and pi independently in (4.2.2), after introducing the constraints (4.2.23), by means of Lagrange multipliers Li (i = 1, ..., n). The modified action integral then becomes
=
1
fl
(4.2.24)
Ll(t, x , x, p , p , A ) dt.
to
It is a standard supposition to assume that the generalized coordinates are specified over a given interval of time, that is, that xi(to)= ai,
xi(tl) =
bi
( a ,bi constants).
158
4. Motion of Conservative and Nonconservative Dynamical Systems
[This supposition was tacitly assumed in the variational formulation of Lagrange's problem, (4.2.21).] Then varying (4.2.24), we easily conclude that the variational equation 61* = 0 is equivalent to the Euler-Lagrange equations :
d aL1 aL1 - 0, dt api api
(4.2.25) (4.2.26)
aL aAi
-=
(4.2.27)
0.
Hence we find, respectively, (4.2.28)
(pi + Ai)'
aH +7 + Ajax
a2H ax' apj
=
0,
(4.2.29) (4.2.30)
Combining (4.2.28) and (4.2.30), we have (4.2.31) For nondegenerative dynamical systems, we must have det(*) api apj # 0,
(4.2.32)
and therefore, from (4.2.31), we find that all Lagrangian multipliers are zero : Ai = 0. (4.2.33) The requirement (4.2.32) is necessary in order that the first group of the canonical equations, .ti= aH/api implicitly define the pi as functions of xi, xi,and t. Thus, li = 0 indicates that even if we ignore the constraint (4.2.23) and vary the action integral (4.2.22), considering x i and pi as independent variables, we will arrive at the canonical differential equations of motion. This shows that the usual assumption of ignoring the contraints implied by
4.2. Hamilton's Canonical Equations of Motion
159
equation (4.2.23) is indeed justified, since the variational equations (4.2.28) and (4.2.29) produce the correct differential equations for Li = 0. Next let us note that the canonical differential equations of motion (and also Lagrangian equations), can be derived from the standpoint of optimal control theory (see [4]). To show this, let us start from Hamilton's action, (4.2.21), and let the generalized velocities X i be nominated as the control variables during the motion of the system, so that 2i
(4.2.34)
= ui
Now, in the terminology of optimal control theory, the action integral (4.2.21) is the cost function, while Equations (4.2.34) can be interpreted as the differential equality constraints. Extremizing the cost function (4.2.21) subject to the constraints (4.2.34) yields the same result as that obtained by extremizing the modified action (see [ 5 ] ) :
I
1
fl
=
[-L(t,X,
to
U)
+ pi(Ui - Xi)] dt,
(4.2.35)
where the pi stand for the Lagrange multipliers. As usual in optimal control theory, we define Hamilton's function to be H(x, U , P , t)
L(t, X , u).
(4.2.36)
[H(t,X , p , U ) - pixi] dt.
(4.2.37)
= pixi -
Thus, the -modified action becomes
I
=
it' to
The variation of I,considering xi,pi and ui as independent variables, yields
(4.2.38)
Assuming that axi = 0 at t = to and t = tl , the condition 61= 0 is equivalent to (4.2.39)
These equations are the equations of motion for a mechanical system and are equivalent to Lagrange's equations and Hamilton's canonical equations. In fact, from (4.2.36), we find that
160
4. Motion of Conservative and Nonconservative Dynamical Systems
and, combining these results with (4.2.39), we obtain (4.2.41)
as well as Lagrange's differential equations of motion, (aL/ax')' aL/ax' = 0. From (4.2.41), we observe that the Lagrange multiplierspi are, in fact, the generalized momenta. The equations of motion (4.2.39) consist of 2n first-order differential equations and n algebraic equations [the third system in (4.2.39)]. In optimal control theory, this algebraic system of equations is called the extremizing condition for the Hamiltonian. Here, this condition yields the relations p; = aL/ax' between the generalized momenta and the generalized velocities. However, such relations were introduced in Hamiltonian mechanics by Legendre's transformation. Finally, let us consider, as a simple illustration, the motion of a heavy particle of mass m constrained to move on the inner surface of a vertical circular cylinder of radius R under the action of a constant gravitational field. At t = 0, the particle is projected in such a way that the initial velocity vector VO makes an angle a with the y-axis (see Figure 4.2.1). Taking the polar angle 8 and vertical distance z for the generalized coordinates, we have x = R cos 8, y = R sin 8, z = z , and kinetic energy of the particle is m m T = - (x2 + 3' + i 2 )= - (R'P + i2). 2 2 Since the potential energy of the particle is
ll = -rngz,
Fig. 4.2.1. Motion of a heavy particle inside a vertical cylinder.
161
4.2. Hamilton’s Canonical Equations of Motion
where g is the acceleration of gravity, the Lagrangian function takes the form L = T - ll
=
m -(R2b2 2
+ i Z+) mgz.
(4.2.42)
The generalized momenta are defined by aL P O= ae = mRZb,
pz = mi.
(4.2.43)
Solving with respect to the generalized velocities, we have the first group of canonical equations : .
1
e = - mR2~e ,
1
i = -Pz.
m
(4.2.44)
The Hamiltonian of the system is defined, according to (4.2.5), to be
H
= p0b
m
+pzi - (R2e2 + i Z) mgz. 2
Substituting (4.2.44) into this relation, we have H=-
1
(4.2.45)
and 6 is a cyclic coordinate. Therefore, the second group of canonical equations is p 0
p z = mg.
= 0,
Hence, p0 = CI,
p z = mgt
+ CZ.
(4.2.46)
Substituting this into (4.2.44), and using the fact that Rb(0) = uo cos a and
i ( 0 ) = v ~ s i n a we , find that
CI = mRvo cos a,
C2 = mvo sin a.
(4.2.47)
Integrating (4.2.44), we finally find that the motion of the particle is given by 8 = (vo/R)tcosa, z = (gt2/2) + votsina, (4.2.48) p0 = mRvo cos a, p z = mgt + mvosina.
162
4. Motion of Conservative and Nonconservative Dynamical Systems
4.3. Integration of Hamilton's Canonical Equations by Means of the Hamilton-Jacobi Method In this section, we focus on Hamilton's canonical differential equations of motion for the case of potential holonomic dynamical systems,
and demonstrate that the problem of integrating these equations can be replaced by an equivalent problem of finding a complete solution of a nonlinear partial differential equation of the first order, called the HamiltonJacobi equation. To show this, we first introduce a scalar field function.
s = S(t, X I , ...,X " ) ,
(4.3.2)
which depends on the time t and the generalized coordinates x i . Let us define the generalized momentum vector of the dynamical system pi to be the gradient of this scalar field function:
as
(4.3.3)
P i = g -
Our aim is to obtain the differential equation that satisfies the field function S(t, x), where x = [ X I , ...,x " ) . Naturally, this differential equation should emanate from the dynamical characteristics of the material system in question. In order to involve the dynamics, we start from the central Lagrange equation, (4.2.1I), considered in Section 4.2, in which we suppose that (a) Qi = 0, and (b) the symbols of variation and differentiation are commutative:
6d - d6
=
0,
(4.3.4)
and that the time is not varied: 6t = 0. Thus, we consider the variations as contemporaneous (simultaneous) Lagrangian variations. Therefore, we have (4.3.5) where L
=
L(t,x,%)is a Lagrangian function for the dynamical system.
163
4.3. Integration of Hamilton's Canonical Equations
Passing to the canonical variables, i.e., using the relation (4.2.5), we have
d
(4.3.6)
(pi 6~')= Li[piXi- H(t, X , P I ] .
Employing the supposition (4.3.3), we find that
;(-$ax')
=
s[ - $ X i
-
..., x",,l,as ...,ax" E)]
H(t,.l,
(4.3.7)
Adding and subtracting the term aWat on the right-hand side, we obtain
From this equation it follows that if we take (4.3.9)
the rest of (4.3.8) becomes the identity
d dt
-(as)
=
ss
which is, of course, compatible with (4.3.4). The partial differential equation (4.3.9) is the Hamilton-Jacobi equation with respect to the field function S(t, x l , ...,x"). In order to interpret the field function in more familiar way, we again use (4.2.5), L(t, X , X) = pixi - H(t, X , p ) . (4.3.10) Adding and subtracting the term as/& on the right-hand side of this equation and using (4.3.3) and (4.3.9), we find that
d W ,x) L(t,x,X) = ___ dt
(4.3.11)
dS = L dt
(4.3.12)
or Therefore, the field function S = j L dt can be interpreted as Hamilton's action integral. Let us suppose that we know a complete solution of the HamiltonJacobi equation (4.3.9), a solution that is an expression of the form
s = S(t,x', ..., X " , c1, ..., C,) + C"+l.
(4.3.13)
164
4. Motion of Conservative and Nonconservative Dynamics1 Systems
where the C's are arbitrary constant parameters. A complete solution of the Hamilton-Jacobi equation is understood to be an expression of the form (4.3.13) that contains the same number of constant parameters Ci as the number of independent variables, t , xl, ...,x", and that, when substituted into (4.3.9), reduces it to an identity. There is another important property of every complete solution of the Hamilton-Jacobi partial differential equation: if we eliminate the constants CI, . . ., Cn+l from (4.3.13) and from the n + 1 equations
pi =
as ax
= f i ( t , X 1..., , x",
~
1
..., , Cn)
(i = 1 ,
..., n), (4.3.14)
as - = F(t, X I , ...,x", CI, ..., Cn), at
we should obtain the Hamilton-Jacobi equation, (4.3.9). However, since the function S does not enter directly into the HamiltonJacobi equation, and since only the partial derivatives of S figure inside it, we can see that one constant should appear additively in the complete solution, as indicated in (4.3.13). Now it is clear also that, for the elimination of C1, ..., C, , and for the purpose of obtaining the Hamilton-Jacobi equation as the result of this elimination, we need only the system (4.3.14). Thus, the expression (4.3.13) is redundant for this purpose. In fact, we will demonstrate that the additional constant C n + l is not an essential parameter in the Hamilton-Jacobi method. Note also that the term aS/at enters linearly in the Hamilton-Jacobi equation. Therefore, in order to obtain the Hamilton-Jacobi equation by elimination of CI ..., C,, in the system (4.3.14), we need only to calculate these parameters from this first group of equations in (4.3.14) and then to substitute them into the second relation in (4.3.14). However in order to be able to calculate CI ..., Cn, we have the condition that the Jacobian determinant be different from zero in the first group of equations in (4.3.14): (4.3.15)
We will now discuss the famous Jacobi theorem, which states that if we have a complete solution of the Hamilton-Jacobi equation of the form (4.3.13), then the solution of the canonical system of the differential equations of motion, (4.3.1), is determined by the following relations:
as
P i = i , ax
as
-=
aCi
Bi
(i = 1,
..., n),
(4.3.16)
4.3. Integration of Hamilton’s Canonical Equations
165
where the Bi (i = 1, ..., n) are a new set of arbitrary constant parameters. The first group of equations in (4.3.16) determine functions of the form pi = Pi(t,x 1 , ...,x “ , C1, ..., Cn).
(4.3.17)
Thus they determine the generalized momenta of the dynamical system in terms of the time t and the generalized coordinates x i . The second group of equations in (4.3.16) give equations of the form
Bi = Bi(t,X1,..., x,, C1, ..., C,)
( i = 1, ..., n).
(4.3.18)
By solving this algebraic system with respect to the x i , we obtain a solution of the dynamical system in the form
where we find the generalized coordinates as functions of time t and the 2n arbitrary constants Ci and Bi. We will also suppose that Equations (4.3.17) are solvable with respect to the parameters Ci, where
Wi(t,X , ..., x n , p l ,...,pn) = Ci 1
(i = 1, ..., n ) .
(4.3.20)
These equations are the conservation laws of Hamilton’s canonical equations (4.3.1). The proof of the Jacobi theorem consists of demonstrating that the solution (4.3.16) satisfies the canonical differential equations of motion (4.3.1) identically. To show this, we start with the Hamilton-Jacobi partial equation (4.3.9), into which we substitute a complete solution to obtain an identity of the form (4.3.21)
The differentiation of (4.3.21) with respect to the various parameters will produce the new identities. First differentiating (4.3.21) partially with respect to the Ci’S, we have
a2s + -aH a2s(t,X , c) = 0
ataci
apj
ax-l aCi
( i , j = 1, ..., n).
(4.3.22)
Taking the total time derivative of the second group of equations in
166
4. Motion of Conservative and Nonconservative Dynamical Systems
(4.3.16), we find that
=0
( i , j = 1, ..., n).
(4.3.23)
These last two systems of equations represent a nonhomogeneous system of linear equations with respect to aH/apj and x’, respectively. The coefficients of these two systems are equal, and, according to (4.3.19, are different from zero. Therefore the roots of both systems have to be identically equal, that is, ..
x’
aH api
-
( i = 1, ..., n).
(4.3.24)
This means that the first group of Hamilton’s canonical differential equations, (4.3. l), are identically satisfied. . To prove that the second group of canonical equations (4.3.1) are identically satisfied, we find the partial derivatives of the identity (4.3.21) with respect to the Xi’s:
a2s(t,x,c) aH a2s(t,x,c) . + i + kJ = 0. at axi ax axi axj
(4.3.25)
We compare this equation with the total time derivative of the first group of equations in (4.3.16): pi = a2s(t,X , c)& azs(t,X , c) axi ad axi at +
(4.3.26)
Substituting (4.3.26) into the identity (4.3.25), we prove that the second group of canonical equations,
aH p i 1 -7, ax
(4.3.27)
is also identically satisfied, which completes the proof of the Jacobi theorem. Therefore, we can conclude that the problem of integration of the canonical equations of motion (4.3.1) can be completely replaced by the problem of finding a complete solution of the Hamilton-Jacobi partial differential equation, (4.3.9). This means that if a complete solution of this equation is available, we can find the motion of the dynamical system without any additional integration by using only the operations of simple partial differentiation and algebra.
4.3. Integration of Hamilton’s Canonical Equations
167
It is of importance to note that there exists an intimate connection between the canonical system (4.3.1) and the Hamilton-Jacobi partial differential equation, a connection that can be traced in the opposite way (direction) from the Jacobi theorem previously stated. In fact, it was also stated by Jacobi that when we know a general solution of the canonical system . aH aH (4.3.29) XI=-, , . - - -=ip api we can always find a complete solution of the Hamilton-Jacobi partial differential equation
at
(4.3.30)
(see, for example, [l]). In other words, finding a general solution the system (4.3.29) and finding a complete solution of the Hamilton-Jacobi partial differential equation (4.3.30) are mathematically equivalent problems. Using the terminology of the theory of first-order partial differential equations, we can say that the ordinary differential equations (4.3.29) are the differential equations of the characteristics of the partial differential equation (4.3.30). It is of interest to point out that in recent years, in many branches of mechanics (especially cellestial mechanics), engineering and optimal control theory, the Hamilton-Jacobi method and the other related field methods have been intensively developed (see, for example, [6] and [7]) and have been applied to many practical problems (see the discussion in the introduction of this Chapter). 4.3.1. Examples
THEHARMONIC OSCILLATOR In order to illustrate the foregoing theory, we consider the elementary problem of a linear harmonic oscillator, namely, a material point of mass m that moves along a straight line (the coordinate x direction) in an elastic field whose elastic coefficient is c. In this case, it is easy to show that Hamilton’s function is 1 1 H = - p 2 + -mo2x2, (4.3.31) 2m 2 where o
=
( ~ / r n ) ’ ’is~ the circular frequency.
168
4. Motion of Conservative and Nonconservative Dynamical Systems
The Hamilton-Jacobi partial differential equation is, therefore, (4.3.32)
To find a complete solution of this equation, consider a function S of the form
s = $X2f(t),
(4.3.33)
where f ( t ) is an unknow function of time. Then the problem is reduced to a Riccati equation: f + (l/m) f + mw2 = 0, whose general solution is f = - mw tan(wt + C ) , where C is a constant. Therefore, a complete solution of (4.3.32) is
S = -tmwx2 tan(wt
+ C ) + D,
(4.3.34)
where D is an arbitrary additive constant that plays no essential role in further considerations. According to the Jacobi theorem, we find that
as
+ C)
p = - = -mwxtan(wt
ax
and
as -_ _
ac -
mwx2 2cos2(wt +
c)= B .
(4.3.35)
(4.3.36)
From the last equation, we find the motion:
x=
(-g) 1/2
cos(wt
+ C).
(4.3.37)
Substituting this into (4.3.39, we can find the generalized momentum: p = - (- 2 m ~ B ) ”sin(wt ~
+ C),
(4.3.38)
which completes the motion of the system. Note also that (4.3.35) implies a conservation law of the form wt
+ arctan(p/mwx)
= const.
(4.3.39)
THE LINEARLY DAMPED OSCILLATOR The linearly damped, one-dimensional harmonic oscillator is governed by the differential equation of motion
R
+ 2kX + wo2x = 0,
(4.3.40)
4.3. Integration of Hamilton's Canonical Equations
169
whose Lagrangian function is L = +(? - o O 2 x 2 p .
(4.3.41)
The canonical momentum is
p =x2kf,
(4.3.42)
and Hamilton's function becomes
H = +(pZe-zk' + &X2e2k').
(4.3.43)
The Hamilton-Jacobi equation thus takes the form (4.3.44)
For a complete solution of this equation, let us seek a function S of the form
~ ( xt ), = +x2f(t)e2k',
(4.3.45)
where f ( t )is to be determined. Substituting this last relation into (4.3.44), we find the ordinary differential equation for f ( t ) :
+f 2 +
f + 2kf
= 0,
(4.3.46)
+ C),
(4.3.47)
which has the general solution
f = -k
-
otan(ot
where C is a constant of integration and
00"
-
k2 =
(4.3.48)
0'.
Thus, the complete solution of (4.3.44) is found to be
S = - i x 2 [ k + o tan(ot
+ C)]2k'.
(4.3.49)
Applying the Jacobi theorem, we have
as
+ o tan(ot + C)]2k'
(4.3SO)
meZk' = B = const., 2 cos2(ot + C )
(4.3.51)
p = - = -x[k ax and
as -- - _ x2 _ ac
which means that
x
=
( - 2 B / o ) 1 ' 2 e - k r ~ ~+~ (Co)t.
(4.3.52)
This is the well-known solution of the linearly underdamped oscillator, (4.3.40). Inserting (4.3.52) into (4.3.50), gives us the generalized momentum
170
4. Motion of Conservative and Nonconservative Dynamical Systems
as a function of time: p
=
k(- 2 ~ / w ) ' / ~cos(wt e~'
+ C ) + (- 2 ~ o ) " ~ sin(ot e ~ ' + C)
(4.3.53)
It is interesting to note that the same problem was treated by Denman and Buch [8] by supposing that the complete solution of the Hamilton-Jacobi equation, (4.3.44), is of the form S ( t , x ) = - Q ( x ) ~ ~In' . this case, the problem is reduced to the differential equation
@y
-
2kQ
+ mix2 = 0,
which cannot be integrated in a closed form. The authors applied an implicit procedure in the application of the Jacobi theorem, which makes the process of finding the motion rather complicated. THEFORCEDOSCILLATOR Let us also consider the forced oscillatory motion of a dynamical system with one degree of freedom, whose differential equation of motion is X
+ O ~ X= h cos a t ,
(4.3.54)
where h, w , and S2 are given constant parameters and w # S2. It is easy to verify that this equation can be interpreted as the Lagrangian equation
for the Lagrangian function of the form L = - x + hS2 sin:;y 2 w2-
-2;
(. -
h cos S2t w2 -
n2
.
(4.3.55)
Introducing the momentum, p = -aL= x
(4.3.56)
ax
the Hamiltonian function becomes +-0
2
2(
x - hCosQr)i. w 2 - a2
(4.3.57)
17 1
4.3. Integration of Hamilton’s Canonical Equations
The canonical equations of motion are then
(4.3.58)
To find a general solution of this system, we turn to the Hamilton-Jacobi equation : o2-
n2
We try to find a complete solution of this partial differential equation of the form (4.3.60)
where f(t) is an unknown function. Substituting this last expression into (4.3.59), we obtain o2= 0 and thus
f=
- 0 tan(ot
+ C).
f +f + (4.3.61)
Therefore the complete solution of (4.3.59) is (4.3.62)
Applying the Jacobi theorem, we find that h cos nt o2-
and
as ac
2
o2-
n2
n2 tan(cot + C ) 1
cos2(ot + C )
= B = const.
(4.3.63)
(4.3.64)
From this last equation, it follows that the motion of the system is cos(0t
+ C ) + o2-h n2cos n t .
(4.3.65)
Taking into account this expression, the momentum (4.3.63) becomes p
= - (-
2 ~ 0 ) sin(cot ” ~
+ C)
and the solution of the canonical system (4.3.58) is complete.
(4.3.66)
172
4. Motion of Conservative and Nonconservative Dynamical Systems
It is of interest to give a brief comment concerning the choice of the Lagrangian function (4.3.55). In fact, as shown in Reference [9], the expression 1 2
- (x
; .(
+ hRsin;;Y o2-
+-0
x-
‘0s
o2-
“7 R2
= D = const.,
(4.3.67)
which differs slightly from the Lagrangian (4.3.59, is a quadratic conservation law of the differential equation, (4.3.54). Thus, the Lagrangian function (4.3.55) was selected by the analogy that exists between the quadratic first integrals of energy type and the Lagrangian functions. Strictly speaking, this analogy is valid only for purely conservative cases but, fortunately, it can be used here by chance. Similarly, for the case of resonance, the differential equation of motion X
+ O’X
=
h cos at
(4.3.68)
admits a quadratic conservation law of the form
’(. Fcoscot 2
-
-
(4.3.69)
It is easy to verify that Lagrange’s equation will produce (4.3.68) for the Lagrangian function
which will be taken as the basis of further consideration. The Hamiltonian function is found to be H = -1p 2 2
+ (ihtcosut + (4.3.71)
and the canonical equations are x=p
p =
h ht + -coswt + -sinot, 2 20
-02’
(
;:
x - -shot
).
(4.3.72)
4.3. Integration of Hamilton’s Canonical Equations
173
The Hamilton-Jacobi partial equation thus becomes
(4.3.73)
whose complete solution is of the form
( Y
y
(
)
S = -to x
-
- sin cot tan(wt + C ) ,
(4.3.74)
where C is a constant. Applying the Jacobi theorem, we find that p = - w x - - ht sinwt tan(wt 20 and
1
+ c)
= B = const.,
(4.3.75)
(4.3.76)
which means that cos(ot
ht + C ) + -sin 20
at.
(4.3.77)
This is the familiar solution for the resonant case. THEFORCED, LINEARLY DAMPEDOSCILLATOR Finally, let us consider the forced, linearly damped oscillator with a single degree of freedom, whose differential equation of motion is X
+ 2kX + O ~ X= h c o s a t .
(4.3.78)
At the same time, consider the Lagrangian function
L = (i[X
+ SZ(A sin SZt - B cos at)]’ - * w i [ x - ( A cos a t + B sin at)]’)2k‘,
(4.3.79)
where A and B are constants to be determined in such a way that the Lagrange equation (d/dt)(aL/aX) - (aL/ax) = 0 generates the correct differential equation of motion, (4.3.78). Inserting (4.3.79) into this Lagrange
174
4. Motion of Conservative and Nonconservative Dynamical Systems
equation, we obtain (x + 2kx
+ o0”x + cosSZt[-A(o0” - a’) - 2kSZBI + sin at[-B(o0” - a’) + 2 k M ] ) 2 k ‘= 0.
(4.3.80)
Therefore, if A and B are selected such that
a’) + 2kSZB = h, 2 k M - B(& - a’) = 0,
A(O: -
(4.3.81)
or A =
(00”
a’) a’)’ + 4k’SZ’ ’
h(& -
B =
(00”
2kha
-
(4.3.82)
a’)’ + 4k’SZ’ ’
then Equation (4.3.80) does, indeed, reduce to (4.3.78). We will now take the Lagrangian function (4.3.79) as the basis for constructing the Hamilton-Jacobi partial differential equation. From Equation (4.3.79), we find Hamilton’s function in the form H = *p2e-2kt- pQ(A sin SZt - B sin at)
+ *o0”[x - (A cos SZt + B sin SZt)]22k‘.
(4.3.83)
The canonical differential equations are then 3 = pe-2k‘ - SZ(A sin SZt - B cos at),
p = - o i [ x - (A cos SZt
+ B sin SZt)]2k‘,
(4.3.84)
where A and B are given by (4.3.82). The Hamilton-Jacobi equation is
+ -21 o z [ x - (A cos SZZt + B sin SZt)]z2k‘ = 0.
(4.3.85)
Let us try to find a complete solution of this equation of the form
S = *[x
-
(A cos SZt
+ B sin SZt)]’eZk‘f(t),
(4.3.86)
where f(t) is an unknown function. Substituting Equation (4.3.86) into (4.3.85), this expression is reduced to f + 2kf + f + 00” = 0. Integrating, we find that
f=
- [k
+ o tan(ot + C ) ]
(C = const.),
(4.3.87)
175
4.4. Separation of Variables in the Hamilton-Jacobi Equation
with (4.3.88)
Therefore,
s = -1 [x - ( A cos SZt + B sin SZt)I2[k+ o tan(ot + C)]e2k'. (4.3.89) Applying the Jacobi theorem, we find the generalized momentum p to be
as
p = - = - [x - ( A cos SZt ax
+ B sin SZt)][k+ o tan(ot + C)]e2k', (4.3.90)
and
- $[x - (A cos Qt
+ B sin Qt)I2cos2(ot + C ) = B = const.
(4.3.91)
From Equation (4.3.91), it follows that x = (- 2 B / ~ ) ' / ~ ecos(ot -~'
+ C ) + A cos SZt + B sin SZt (4.3.92)
which is the well-known solution of (4.3.78). Inserting this result into (4.3.91), we find the momentum as a function of time: p = - ( - 2 B / o ) ' / 2 e k t [ osin(wt
+ C ) + kcos(ot + C ) ] ,
(4.3.93)
where B and C are arbitrary constants.
4.4. Separation of Variables in the Hamilton-Jacobi Equation
Although it is not possible to find any universal method for constructing a complete solution of the Hamilton-Jacobi partial differential equation, there are a number of cases in which finding a complete solution is straightforward; in such cases, obtaining the motion of a dynamical system is simpler by the Hamilton-Jacobi method that integrating the differential equations of motion directly. Very frequently, under certain conditions, it is possible to separate the variables of the Hamilton-Jacobi equation and the problem is reduced to quadrature. For example, one of the commonIy occurring cases where finding a complete solution is considerably simplified is the case of a purely
176
4. Motion of Conservative and Nonconservative Dynamical Systems
conservative dynamical system, for which Hamilton's function does not depend explicitly on time t. The Hamilton-Jacobi equation then becomes
s, ...,
(4.4.1)
+ Si(X', ...,x"),
(4.4.2)
..., x",
at
By introducing the transformation
S = -Et
where E is a constant, Equation (4.4.1) reduces to the form (4.4.3)
It is clear that the constant E has the interpretation of total mechanical energy of the dynamical system [see (4.2.18)]. Suppose we find a complete solution S of the partial differential equation (4.4.3), which, in addition to x ' , ...,x" contains the constant CI = E and (n - 1) constants CZ,..., Cn :
5'1
= Sl(x',
Then (4.4.2) becomes S = -Et
..., x", E, C2, ..., Cn).
(4.4.4)
+ S l ( X ' , ...,x",E, C2, ..., Cn).
(4.4.5)
Applying the Jacobi theorem, (4.3.16), we have that the motion of the dynamical system is determined by the relations
as
-=
aE
- t + -asl =B aE
1
- -to,
as, ac,
-=
B,,
a = 2,3,
..., n.
(4.4.7)
Next, if some of the generalized coordinates are cyclic (ignorable), then the method of separation of variables can be advantageously applied. Let us consider the case for which n - k independent coordinates are ignorable, in that they do not appear explicitly in H . The Hamilton-Jacobi equation is of the form
as
,...,x k , s ,...,
(4.4.8)
The complete solution of this equation can be expressed as a linear function relative to all of the ignorable coordinates, namely, S = s ~ ( x ' , ..,x k , ~
1
..,., Cn, t ) + C ~ X ,
(a = /t
+ 1, ...,n).
(4.4.9)
4.4. Separation of Variables in the Hamilton-Jacobi Equation
177
From this equation we have
as -- -as2 _ at
as
at
-=
ax,
’
C,
as
as,
ax1
axi
.-
(a = k
(i = 1,2, ...)k);
+ 1, . . . , T I ) .
(4.4.10) (4.4.11)
From (4.4.11), on the basis of (4.3.16), it follows that n - k momenta are constants. Therefore, the Hamilton-Jacobi equation (4.4.8) becomes
If a complete solution of this partial differential equation is available, then by applying the Jacobi theorem, we can easily find the solution of the corresponding dynamical system. Finally, let us consider separation of variables for dynamical systems whose Hamiltonian functions are of the form H = H [ t ,X I ,
...,x k ,PI, ...,Pk,fk+l(Xk+*,pk+l),-. ,fn(x”,Pn)],
(4.4.13)
depends only on the corresponding single pair where every function (couple) of canonical variables x i , pi. The Hamilton-Jacobi equation corresponding to (4.4.13) is
=o
(4.4.14)
For a complete solution of this equation, we seek a function S of the form S = V(X’,...,x k , C1, ..., Cn, t )
n
+ C
j=k+l
W(X’, C1,
- 9
Cn), (4.4.15)
where wj is the additive part of the action depending only on the generalized coordinates XJ and all constant parameters Ci(i = 1 , ...,n). From (4.4.15), we have
as -- av _ at at ’ -as -
ax,
- -aw, ax,
as
av
a x i - axi (a = k
(i = 1 ,
+ 1, ..., n),
...,k),
(4.4.16) (4.4.17)
178
4. Motion of Conservative and Nonconservative Dynamical Systems
and the partial differential equation (4.4.14) becomes
av
t , X 1 , ..., x k - - . - 7axk "",fk+l(x
at =
k+l
Y
awk+l
**-,f"(x"7$)]
(4.4.18)
0.
If the expression (4.4.15) is indeed a complete solution of the HamiltonJacobi equation (4.4.14), then the partial differential equation (4.4.18) should become an identity for arbitrary values of the generalized coordinates x k + ' , ...,x". However, this is possible only under the condition that all functions fi ( x i , a W / a x ' ) ( i = k + 1, ..., n) have a constant value: (a = k
+ 1, ..., n).
(4.4.19)
This is a system of n - k ordinary differential equations. Now, the Hamilton-Jacobi equation, (4.4.18), becomes -
at
+H
(
XI,
av ...,X k Y
av aXk'
If we are able to find a complete integral of this partial differential equation and to integrate (4.4.19), then, using (4.4.16), (4.4.17), and the Jacobi theorem, we can find the solution of (4.4.14) in the form Pi =
av a wa
Pa = axa
as _ - Bi aCi
( i = 1,
...)k),
(a = k
+ 1, ..., n),
(4.4.21)
(i = 1, ..., n).
The method of separation of variables (coordinates) we have just considered includes the ignorable coordinates as a special case. In fact, by putting fO1(xa,pa) = p a (a = k + 1, ...,n) in (4.4.13), the system (4.4.19) becomes
a wa - Ca
-ax.
(a = k
+ 1, ..., n),
(4.4.22)
and the complete integral (4.4.15) is identical with the complete solution (4.4.9) considered before. The separation of variables in the Hamilton-Jacobi theory can be considered, in some sense, to be the generalization of the notion of cyclic
4.4. Separation of Variables in the Hamilton-Jacobi Equation
179
(ignorable) coordinates and their conservation laws. Also, the possibility of performing the separation of coordinates depends on the coordinate system (it is a privilege of the coordinate system) in which the motion is described. As a simple illustration, consider the motion of a heavy particle on the inner surface of a vertical cylinder, which we considered previously in Section 4.2. Since Hamilton’s function is given by (4.2.45)’ the HamiltonJacobi partial differential equation is of the form
(-)az
a~ + L(57 + 1 as at
2mR2
ae
2m
- mgz
=
0.
(4.4.23)
Since the Hamiltonian does not depend on time t explicitly and 9 is a cyclic coordinate, we seek a complete solution of Equation (4.4.23) of the form
S = -Et
+ A9 + f ( z ) .
(4.4.24)
Substituting this into (4.4.23), we find that
f
= 2m2gz - ( A / R )
+ 2mE.
(4.4.25)
Integrating, we obtain the complete solution of the form S = -Et
+A8 +
+
Applying the Jacobi theorem, it follows that the generalized momenta are given by pe = A , p z = A[2m2gz - (A/R)’ + 2mE]1/2, (4.4.27) and 1/2 as - 8 - 2A _ 2mZgz + 2 m ] = B1 = const., aA 3m2gR2
67 67 +
[
as -- - t _ aE
+
mg
[
2m2gz -
2m]
= B2 =
const. (4.4.28)
According to the given initial conditions, for t = 0 we have
B
= 0,
z = 0,
pe = mRvo cos a ,
pz
=
mvo sin a. (4.4.29)
Therefore, A = mRvo cos a, vo
BZ = -sina, g
2 vt sin a cos a B1 = - ’ 3 SR
mvt E =-
2 .
(4.4.30)
180
4. Motion of Conservative and Nonconservative Dynamical Systems
Using these relations, and eliminating the term [2m2gz - ( L ~ / Z ? ) ~+ 2mE]”z from (4.4.28), we obtain
e = (uom)t cos a Also, from the second equation in (4.4.28), we find that z = gt2 - + uot sin a. 2 Finally, putting this into the expression for p z , given by (4.4.27), we get pz
=
m(gt
+ uosina),
which completes the solution.
4.5. Application of the Hamilton-Jacobi Method to Linear Nonconservative Oscillatory Systems The method of Hamilton and Jacobi is formulated for potential dynamical systems, namely, those whose behavior is completely described by a Lagrangian or Hamiltonian function. As underlined in the introduction of this chapter, many important problems arising in physics, mechanics and engineering have been solved by means of this powerful method. In this section we demonstrate that under some conditions, the HamiltonJacobi method can be applied to a purely nonconservative linear system that does not possess a Lagrangian function. Let us consider a linear dissipative dynamical system whose differential equations of motion are of the form
xi + b.. rjxj’. + w 2Xi
=0
( i , j = 1,
..., n).
(4.5.1)
Conservation laws for this type of dynamical system were studied in Section 3.21. Note that subscripts on the variables are equivalent to the previous superscripts. In order to incorporate gyroscopic forces (whose tensor is antisymmetric: bij = -bji) and dissipative forces (whose tensor is, as a rule, symmetric: bij = bji), we will suppose that the coefficients bij in (4.5.1) are in general not symmetric; that bij # bji. Since the differential equations of motion, (4.5.1), are not derivable from a Lagrangian function, we turn to the possibility that some (combinations) of these equations can be described by means of a Lagrangian function.
4.5. Application of the Hamilton-Jacobi Method
181
Let us take the first differential equation of (4.5.1) as it stands and multiply the second by a constant factor 1 2 , and the third by h3, etc., and then sum all of these equations. Thus, we obtain, for a = 2,3, ...,n,
+ 02(x1+ 2,x,)
(4.5.3)
=0
We introduce a new variable, Y = x1 + Lax,.
Let us select the multipliers 1, (a = 2, 3,
bl2 611
+ 1,b,2
+ 1,bd
= 12,
(4.5.4)
...,n) in such a way that
..., bbll~ n++ Axban = 1,. 1,bd
(4.5.5)
Then, denoting by
2k
=
bll
+ Aab,l,
(4.5.6)
we reduce Equation (4.5.3) to the form Y+2kY+u2Y=Oy
(4.5.7)
which can be derived by means of a Hamiltonian or Lagrangian. By using Equation (4.5.6), we can write the system (4.5.5)in the form of n - 1 homogeneous algebraic equations [see also (3.21.19)]:
If we solve this system with respect 1, and substitute the solution into
182
4. Motion of Conservative and Nonconservative Dynamical Systems
(4.5.6), we obtain an nth-order determinantal equation,
bii - 2k
biz
b21
bzz - 2k
bn i
bn2
.
...
bln
*.*
b2n
=
0,
(4.5.9)
bnn - 2k
as the requirement for a nontrivial solution with respect to 2k. Solving (4.5.9) by supposing that there are no multiple roots, we obtain 2k(l), ...,2k(n). Substituting these into (4.5.8) and solving, we find & ( I ) , & 4 2 ) , ...,.Aa(n) (a = 2,3, . .., n). Therefore, Equations (4.5.4), (4.5.6), and (4.5.7) are in fact
and
qi)= X I + 1 2 a ( ; ) ~ a (i = 1 , ..., n, a = 2,3, ..., n), 2k(i) = bil + L(i)bal,
ci)+ 2k(;)$i) +
o2Ki)=
0.
(4.5.10) (4.5.11) (4.5.12)
In the last equation, and in the subsequent text, the summation convention with respect to the repeated indices in the parentheses ( ) is not assumed, while this convention is applied to all other indices that are not in parentheses. As pointed out already, the system of equations (4.5.12) admits a Lagrangian function of the form '
n
Hamilton's function thus becomes
where the generalized momenta are defined by (4.5.15)
The Hamilton-Jacobi partial differential equation becomes
To solve this equation, we apply the method of separation of variables, that
4.5. Application of the Hamilton-Jacobi Method
183
is, we suppose that
S(t, & I ) ,
..-,Vn))
+ ... + S(n)(t, qn)).
= S(l)(t,ql))+ S(z)(t,q ~ ) )
(4.5.17)
Substituting this into (4.5.16), we get n equations, one for each of the q;,,t ):
S(i)(
For each of the n degrees of freedom, we are led to equations that are completely analogous to (4.3.44). Therefore, by analogy with (4.3.49), we get ~ ( i= )
-
3~ & [ k ( +i ) o tan(ot + ~ ( i ) ) ] e ~ ~ ( ~ ) '(4.5.19) ,
where the C(i)[ ( i ) = 1, ...,n] are constants. Applying the Jacobi theorem, we have
and
From this last group of equations, it follows that
qi)= [-
~B(~)/CO]'/~~-'(I'' COS(CO~ + C(i)).
(4.5.22)
Using (4.5.22), the generalized momenta, (4.5.20), can be expressed as functions of time: = [k(i)(-2B(i)/o)'''
COS(CO~ + C(i,) + (- 2B(i)a)'/' sin(ot
+ C(i))]ek(O'. (4.5.23)
Let us suppose that the initial conditions for the system (4.5.1) are given as bi. Using (4.5.10) and (4.5.15), we can find the initial values qi)(O) andpi(0). Therefore, from (4.5.22) and (4.5.23), we can find the 2n constants B(i) and C(i)in terms of ai and bi. The motion of the dynamical system is given by (4.5.10) and we suppose that this expression can be inverted to give Xi = xi(t). The method of Hamilton-Jacobi can also be applied to dynamical problems whose differential equations of motion are of the form
x;(O) = ai and Xi(0) =
2i
+ 2kki + aijxj = 0
(i, j = 1, ...,n).
(4.5.24)
Conservation laws for this type of dynamical system are discussed in Section (3.20).
184
4. Motion of Conservative and Nonconservative Dynamical Systems
In order to include nonconservative position forces into consideration, we suppose that the coefficients aij are not symmetric: a0 # aji . Therefore, the dynamical system is not conservative and a Lagrangian function cannot necessarily be found. However, repeating the same steps as in the previous case, namely, multiplying the second equation (4.5.24) by a constant factor p2, the third by p3 etc., and summing all equations, including the first, we obtain, for a = 2 , 3 ,..., n,
+
where X
= x1
Denoting by
+ (sin + p,aan)Xn = 0,
*.*
+ pax,
(a!
2,3,
=
(4.5.25)
..., n).
(4.5.26) (4.5.27)
and selecting p, in such a way that a12 all
+p
+ p,a,l
~ -2 - p2,
we reduce (4.5.25) to
ain
+ paaan = p n ,
..., all + p,a,l
+ 2 k Z + 0 2 X = 0.
X
To determine o2and the unknown multipliers p, use (4.5.27) to express the system (4.5.28) as
+
pz(a22 - 0 2 ) p3 a32
+
+
~ 2 a 2 ~ ~ 3 a 3 ~
(4.5.29) (a!
+ - - - + pnan2
+ pn(ann
(4.5.28)
=
=
- 02) =
2, 3 ...,n), we first
- a12
-aln.
Solving this system with respect to p,, we substitute the result into (4.5.27) and, after a simple transformation, we obtain the determinantal equation all
- o2 a21
an1
ain
a12
a22
azn
- o2
an2
(Inn
-
= 0,
(4.5.31)
o2
which is an algebraic equation of the nth order with respect to
02.
4.5. Application of the Hamilton-Jacobi Method
185
Therefore, from (4.5.31) and (4.5.30), we find o t l ) , ..., o t n ) and ...,p,(n) [assuming that (4.5.31) has distinct roots]. From (4.5.26), (4.5.27), and (4.5.29), we actually have
p,(l),
X(i)
= XI
+ p a ( i ) X , (a = 2, 3, ...,n, (i) = 1, ..., n), 4) = a l l + p,(i)a,i, 2 ( i ) + 2 k ~ ( i+ ) o h ) X ( i ) = 0,
(4.5.32) (4.5.33) (4.5.34)
where the repeated indices in the round brackets (i) are not summed. Since the system of transformed differential equations (4.5.34) is derivable from a Lagrangian function of the form n
L =
C +[x& - W $ ) X , $ ) I ~ ~ ~ ~ , i=l
(4.5.35)
we can, repeating the same procedure as outlined in the previous case, apply the Hamilton-Jacobi method. For purely conservative dynamical systems whose Lagrangian function is L
= La..%. 2 IJ IXJ .. - 1 ~CijXiXj,
(4.5.36)
where a i j = a j i and Cij = C j i , the method suggested in this section, which is based on the introduction of undetermined multipliers A,, is identical to the method of principal (normal) coordinates in linear vibration theory. However, the method described here, in contrast to the method of normal coordinates, does not require the knowledge of any Lagrangian function and can be applied, as we demonstrated, to nonconservative cases that do not possess Lagrangian functions. Frequently, we can apply the method of multipliers to forced vibration problems, which will be demonstrated by means of the following example. 4.5.1. The Linear Forced Vibration System with Two Degrees of Freedom As a practical example, one that serves as an illustration of the foregoing general considerations, let us study the motion of a linear forced system with two degrees of freedom, whose differential equations of motion are of the form a l l x l + ~ 1 2 x 2+ c l l x l + ~ 1 2 x 2= H I sin(pt + 4, (4.5.37) ~ 1 2 x 1+ a 2 2 A + ~ 1 2 x 1+ ~ 2 2 x 2= Hzsin(pt + a),
where a i j = parameters.
aji,
Cij = C j i ,
and H I , H2, p, and 6 are given constant
186
4. Motion of Conservative and Nonconservative Dynamical Systems
Multiplying the second equation by a constant multiplier A and adding it to the first, we find that (all
+ A a d X 1 + (a12 + Aa22)x2 + (c11 + Ac12)xl
+ ( c 1 2 + A c 2 2 ) x z = ( H I + AHz)sin(pt + 6)
(4.5.38)
or (a12
+ Aa22)
all a12
+ Aa12 + La22
c11 c12
= (HI
Dividing by
a12
+ Ac12
+ Acz2 x 1 + x 2
+ AH2)sin(pt + 6)
(4.5.39)
+ A a 2 2 and denoting by (4.5.40)
it is clear that under the condition (4.5.41)
we can write the differential equation (4.5.39) in the form of the following system of differential equations :
where the repeated (i) is not summed and quadratic equation A2 -
a22c11 a12c22
- a11c22 A+ - a22c12
A(1)
and
=
+ A ( i ) a i 2 x1 + x 2 a12 + A(i)a22 all
= K(i)Xl
+ xz
are the roots of the
a11c12
-
~ 1 2 ~ 1 1
a12c22
-
a22c12
which follows from (4.5.41). In Equation (4.5.42), X(i)
4 2 )
=
=
0,
X ( i ) denotes
(4.5.43)
the quantity
+ x1 + x2 1 + 2 A(i)~22
~ 1 1 A(ilc12 ~
[(i) = 1,2].
(4.5.44)
Finding A 1 and A 2 from (4.5.43), we also find the squares of circular frequencies [see (4.5.40)] : (4.5.45)
4.5. Application of the Hamilton-Jacobi Method
187
Before we apply the Hamilton-Jacobi method, it is of interest to find a more explicit form of the parameter K , which was introduced in Equation (4.5.41).According to (4.5.41),A(i) and K(i) are connected in the following way:
From these equations, we find that - a11c22 - K(i) = a22c11 .
A(i)
-K(. A . ’)
=
(I)
a12c22
- a22c12
allCl2
- alZCll
a12c22
- a22c12
Y
(4.5.47)
-
However, from these expressions, it follows that for A(1)
=
- K(z)
and
A(z) = -Kw,
(4.5.48)
we have A(1)
+ 42)
A(l)A(2)
= =
a22c11
- a11c22
a12c22
- a22c12
UllC12
- U12Cll
a12c22
- azzc12
9
(4.5.49)
Hence A(1) and A( 2) are the roots of the quadratic equation (4.5.43). In order to incorporate the differential equations (4.5.42) into a Lagrangian problem suitable for application of the Hamilton-Jacobi method, we try to find a Lagrangian function of the form
L =
i[z&- Alpcos(pt + 41’
+ 41’ + +[k&- A2p cos(pt + 41’ - +[ X( Z)- A Zsin(pt + a)]’, -
o&[X(1)- A1 sin(pt
(4.5.50)
where A1 and A Z are constant parameters to be determined in such a way that the Lagrange equations,
(4.5.51) generate the correct differential equations, (4.5.42). Substituting (4.5.50) into (4.5.51),we obtain
188
4. Motion of Conservative and Nonconservative Dynamical Systems
Comparing these equations with (4.5.42), we find that A1 =
Hi
(a12
+ A1H2
+ Alazz)(w?l) - p2)'
(4.5.53)
Hi + A 2 H 2 A2 = (a12 + A2(122)(0?2)- P2) * Introducing the generalized momenta by the relations
(4.5.54)
the Hamiltonian function is found to be H = '2 2 P1
+ AlPlpCOS(pt + 6) + AzPzpcos(pt + s)
+
+ &?i)[X(1)
- A1 sin(pt
+ 6)12 + +o&[X(2)- A2 sin(pt + )I2
(4.5.55)
Therefore the Hamilton-Jacobi partial differential equation is
+- as
A2p cos(pt
aX(2)
+ s)+ +o&[X(1) - A1 sin(pt + @I2
+ +o?z)[X(~) - A2 sin(pt + s)I2
= 0.
(4.5.56)
We seek a complete solution of this partial differential equation, of the form S = ~ [ X ( I-) A1 sin(pt
+ i5)I2f1(t) + +[X(Z)- A2 sin(pt + 6)12f2(t),
(4.5.57)
where fl(t) and f 4 t ) are unknown functions of time. Substituting (4.5.57) into (4.5.56), we obtain f1 + fi" + ot1) = 0 and f2 + fi" + o("z) = 0. Integrating, we find f1 = - o(1)tan(o(1)t + Cl) and f2 = -w(2) tan(o(2)t + C2), where C1 and C2 are arbitrary constants. Therefore, the complete solution of (4.5.56) is S = -&o(l)[X(l) - A1 sin(pt
-+[X(2) - A2sin(pt
+ S)I2tan(o(1)t + CI)
+ ~3)]~0(2) tan(o(2)t + C2).
(4.5.58)
189
4.5. Application of the Hamilton-Jacobi Method
Applying the Jacobi theorem, we get the generalized momenta:
as
pl=--
ax,,
-
-o(l)[X(l) - A1 sin(pt
+ 4 1 tan(w(1)t +
as p 2 = - - - -O(Z)[X(Z) - A Z sin(pt + S)l tan(o(z)t +
Cl),
(4.5.59) CZ),
and
where B1 and BZ are arbitrary constants. The last two equations can be solved with respect to the normal coordinates
+ Cl) + A1 sin(pt + 6), X ( Z )= (- ~Bz/w(z))'/~ cos(w(gt + CZ)+ A Zsin(pt + a), X(1) = (- 2B1/0(1))'/~ cos(o(1)t
(4.5.61)
where A1 and A2 are given by (4.5.53). However, in accordance with (4.5.44) and (4.5.48), X(1) = -A(z)X1 4 2 )
Solving for XI and
XZ,
= -A(l)Xl
+ x29 + x2
(4.5.62)
*
we have
Substituting (4.5.61) into (4.5.63), it follows that
+ Cl) + DZcos(o(~)t+ CZ)+ R1 sin@ + 6), xz = 1(1)D1cos(o(1)t + Cl) + L(Z)DZ cos(w(2)t + C Z )+ RZsin(pt + 6) XI
= D1 cos(o(~)t
(4.5.64)
where
190
4. Motion of Conservative and Nonconservative Dynamical Systems
(4.5.65)
-
-
P2)
42)(Hl (a12
+ L(Z,H2)
+ 42)a22)(0?2) - P2)
Note that using (4.5.49) for A = A(1) and 1 = 4 2 ) , as well as (4.5.48), (4.5.49, and (4.5.46), we can (after laborious but simple algebraic manipulations) transform the last two expressions to the following form: R1 = R2 =
- U 2 2 P 2 ) - H2(c12 - U 1 2 P 2 ) (a11a22 - a?2)(0?1,- P2)(0?2) - p 2 )’ Hl(C22
a11p2) - H l ( C l 2 - a12p2) - a?2)(~?1) - p2)(wt2)- p 2 ) ’
HZ(Cl1 (ma22
(4.5.66)
4.6. A Field Method for Nonconservative Dynamical Systems
As briefly demonstrated in the previous sections, the Hamilton-Jacobi method can be advantageously used in many practical situations as an exact method for solving the canonical differential equations of motion. In addition, a variety of approximate procedures can be built up, based on this method, for solving the nonlinear problems for which an exact, complete solution of the Hamilton-Jacobi partial differential equation is not available. For example, the interested reader will find an exhaustive review of applications of the Hamilton-Jacobi method within the realm of nonlinear vibration theory and celestial mechanics in the monographs of Kevorkian and Cole [lo] and Nayfeh [ll]. However, the method of Hamilton and Jacobi can be applied only to those systems that are completely described by the LagrangiadHamiltonian function, and the purely nonconservative dynamical systems stay outside of the areas treated by this method. In this and the following sections of this chapter, we discuss a field method that is suitable for studying the motion of purely nonconservative dynamical systems. The method presented here can be also applied to conservative dynamical systems and to those nonconservative systems that are
4.6. A Field Method for Nonconservative Dynamics1 Systems
191
generated from a LagrangiadHamiltonian. In this respect, the field method presented here differs completely from the Hamilton-Jacobi method. Let us consider a nonconservative, holonomic, rheonomic, dynamical system whose differential equations of motion are of the form fl
=
Xl(t, x1,
...,xn), (4.6.1)
fn
= Xn(t, x1,
...,x n ) .
Note, that by the term “nonconservative,” we understand that the number n may not be an even number; also, we do not insist that we be able to write the system (4.6.1) in the canonical form (4.2.20). The basic supposition of our theory is based on the hypothesis that one coordinate of the system, say X I ,can be selected as the basicfield, depending on time t and the rest of the coordinates describing the dynamical system: x1
=
cp(t,x2,
...,xn).
(4.6.2)
Differentiating this expression totally with respect to time and using the last n - 1 equations of (4.6.1), we write the first differential equation of this system in the form acp
at
acp + -X,(t, ax, = O
cp,x2,
...,xn) - Xl(t, cp,x2, ...,x n )
( a = 2 , 3 ,..., n),
(4.6.3)
where the summation convention with respect to repeated indices is applied. We call the partial differential equation (4.6.3) the basic field equation. It plays the same role in our considerations as the Hamilton-Jacobi partial differential equation in the previous discussion. However, it should be noted that, in contrast to the Hamilton-Jacobi partial differential equation, which is, as a rule, strictly nonlinear, the basic field equation (4.6.3) is a quasilinear partial differential equation and, therefore, has a simpler structure than the Hamilton-Jacobi equation. That is to say, the solution of the basic field equation (4.6.3) forms a hypersurface in the n-dimensional space t, x 2 , ...,xn . According to the geometric theory of quasi-linear partial differential equations (see, for example, [l]), this hypersurface is generated by the family of characteristic curves whose differential equations are given by the system of ordinary differential equations (4.6.1). In general, the geometric properties of the integral surface O(t,xz , ...,xn)are considerably simpler than the geometrical properties of the integral surface S(t,X I , ...,xn) of the corresponding Hamilton-Jacobi theory, since the gradients aS/ax’ appear nonlinearly in the Hamilton-Jacobi partial differential equation.
192
4. Motion of Conservative and Nonconservative Dynamical Systems
In order to demonstrate certain variational properties of the basic field equation, we consider the differential equations of motion in the nonconservative Hamiltonian form (4.2.17), namely,
where x i (i = 1, ...,n) are the generalized coordinates and pi are the generalized momenta. The Qi are purely nonconservative forces. As shown in Section 4.2, the equations of motion can be related to the central Lagrangian equation, (4.2.12), which under the supposition that 6d - d6 = 0, has the form d -(pi 6 ~ '-) 6(piki - H ) - Qi 6 ~ =' 0 dt
(i = 1, ..., n). (4.6.5)
By adding and subtracting the time derivative of the quantity Udx' in the last equation, where U is an arbitrary function depending on time t , on the generalized coordinates x l , ..., xn, and on the n - 1 components of the momentum vector p2 = ZZ,...,pn = Z n , we find that, for i = 1, ...,n, d d z ( p i S X i - U ~ X '-) 6(pi6Xi - H ) + Qi6Xi + - ( U ~ X ' )= 0. (4.6.8) dt
At this point, we define the first component of the momentum vector p1 to be equal to the field function U : p1 = U(t,X1,...,~
...,
" ~ 2 2 , zn).
(4.6.7)
Therefore, we can write (4.6.6) in the form d dt
-(z, SX,) - 6(UX'
d + z,Xm - H ) - Qi 6 ~ -' Q,6 ~ +" dt - ( U ~ X ' )= 0
(4.6.8)
for 01 = 2,3, ..., n. Writing this equation explicitly, and taking into account that dx', ax", and 62, are independent, we obtain the following system of equations :
(4.6.9)
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
193
Calculating i, and i afrom the last two expressions and inserting them into the first equation of the system (4.6.9), we find the following basic field equation for the system (4.6.4):
=0
(a! =
2 , 3 , ...,n)
(4.6.10)
in which, according to (4.6.7), the basic field is taken to be the first component of the momentum vector. As indicated previously, we can derive the basic field equation (4.6.10) by substituting the field (4.6.7) into the equation p1 = - ( a H / a x l )
+ Q~
(4.6.11)
and using all of the rest of the equations of the system (4.6.4). However, the derivation given here is of mechanical significance, since it reveals the invariant character of the basic field equation, which is based on the central Lagrangian equation, i.e., D'Alembert's principle. The field approach proposed here was introduced in References [ 121 through [ 141. Naturally, the set of differential equations (4.6.1) and (4.6.4) are fully equivalent for the majority of the dynamical systems. However, in the text that follows, we will consider only one type of system, without any loss of generality.
4.7. The Complete Solutions of the Basic Field Equations and Their Properties In what follows, we shall be concerned with the complete solutions of the basic partial differential equation (4.6.10). We define a complete solution to be a relation of the form p1 =
U(t,X1,..., ~
...,
" ~ 2 2 , ~ n C1, ,
..., Czn);
(4.7.1)
in addition to the arguments t , x', ..., x", 2 2 , ..., z,, , the solution p1 contains 2n arbitrary constants and, when substituted into Equation (4.6.lo), reduces it to an identity. Note that every complete solution satisfies the property mentioned in Section 4.3; that is, if we eliminate the 2n constant parameters C1,...,Czn from the 2n + 1 relations (4.7.1), aU/at, aU/axi, and aU/az, ( i = 1, ..., n, a! = 2 , 3 , ..., n), we should obtain the basic field equation (4.6.10).
194
4. Motion of Conservative and Nonconservative Dynamical Systems
4.7.1. Bundles of Conservation Laws Several observations can be made. With respect to the differential equations of motion (4.6.4), the expression (4.7.1) represents a bundle of conservation laws for the dynamical system. That is to say, we can recover all of the 2n conservation laws of the dynamical system (4.6.4) from the complete solution (4.7.1) of the basic field equation by giving any particular values to the 2n - 1 constants Ci, in their relevant domain, and by allowing one of them to be arbitrary. For example, the set of relations
p1 = Ul(t,X1,..., xfl,22,..., Z n , cl),
p1
=
u2n(t,x1,..., x ~ , z..., ~ ,~
=
~2
~1
n c Z, ~ ) ,
~3
=
=
= CZ =
~ 2 = ,
0.
= ~ 2 n - 1=
0.
represents a complete set of conservation laws for the dynamical system (4.6.4) under the condition that all of these relations are mutually independent. In other words, under the condition that aUi/aCi # 0 (i = 1,2, ...,2n), we can invert (4.7.2) to obtain the more familiar form for the conservation laws:
@l(t,x l , ..., x n ,21, @2n(t,x l , ...,X ' ,
..., 2,)
=
c1, (4.7.3)
21
. . a ,
2,) = ~ 2 n
(zi
pi).
4.7.2. Initial Conditions and Conditioned Forms The bundle of conservation laws (4.7.1) can also be matched with the initial conditions of the dynamical system, if they are formulated as an initialvalue problem. Let the initial conditions of the system (4.6.4) be given by
~'(0) = a',
pi(O) = bi,
(4.7.4)
where a' and bi are given constants. Applying (4.7.4) to (4.7.1) and expressing one constant, say CI,in terms of a', bi, and the rest of the parameters C2, ..., C z n , we can write (4.7.1) in the following conditioned form: p1 = O(t,x', ..., x n , 2 2 , ...,Z,, a1, ...,an,b l , ...,b,, c2, ..., C2,).
(4.7.5)
It should be noted that it is a characteristic feature of the theory that the constant parameters C2, ..., C2n in the conditioned-form solution, (4.7.9, remain undetermined through the entire process of finding the motion.
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
195
We will now demonstrate that the motion of the dynamical system, which moves in accordance with the differential equations of motion (4.6.4) and the initial conditions (4.7.4), can be obtained from the expression (4.7.5) and the 2n - 1 algebraic equations
-a -0 - 0
( A = 2 , 3 , ..., 2n)
acA
(4.7.6)
for arbitrary values of the constants CA. We suppose that the following functional determinant:
a20 ... a2O a20 ... ___ ac2ax1 ac2axn ac2az2
~
~
a20 ~
aC2
az, (4.7.7)
a20
ac2,ax'
...
a20
a20
aC2, ax"
aC2,az2
...
a20 aC2,
az,
is nowhere zero in the relevant domain of xi, z, and CA (i = 1, ..., n, (Y = 2 , 3 , ..., n, A = 2, 3 , ..., 2n). It is clear that the requirement (4.7.7) can be interpreted as the condition of solvability of the 2n - 1 Equations (4.7.6) with respect to the 2n - 1 quantities xi and zol.However, we will suppose that Equations (4.7.6) are not solvable with respect to the CA, that is, they become an identity for t = 0, for arbitrary values of CA. To prove the previous statement, we first differentiate Equation (4.7.6) totally with respect to time:
a20 x i + a2ii 2, +acAat acAa2 acAaz, a20
~
( A = 2 , 3 , ..., 2n, i
=
1,2,
(4.7.8)
=0
...,n, a = 2 , 3 , ..., n).
On the other hand, since the right-hand side of (4.7.5) satisfies the basic field equation (4.6.10) identically, we can form a partial derivative of the basic field equation (4.6.10) with respect to CA, to obtain.
a20
a20 a H
a20 a H
+
a2H [-- a 0 2 +--axwaz,ao + aOa2H
ax'
aO
az,
a20
aH
L au (n.- g) (4.7.9)
196
4. Motion of Conservative and Nonconservative Dynamical Systems
Employing (4.7.6) and forming the difference between (4.7.8) and (4.7.9), we obtain
+ 5(2, +
acAaz,
(A = 2 , 3 ,..., 2 n , a = 2 , 3 ,..., n). This is a set of linear algebraic equations that is homogeneous in the quantities
Since the determinant (4.7.7) is different from zero by hypothesis, it follows that . aH aH X' = ( a = 2 , 3 ,..., n , i = l , ..., n) a n ' i , = p , = - - +ax.Q , (4.7.1 1) which are the 2n = 1 canonical equations (4.6.4). In order to show that the first canonical equation,
aH
P I = --
ax' +
(4.7.12)
Qi,
is also satisfied, we form the time derivative of (4.7.5) and combine the resulting equation with (4.7.11):
au
auaH ax, az,
. auaH pi = - + - - +--+-
at
ax'au
au
az,
( ax..
aH
)
--+Qa.
Substituting this into the basic field equation (4.6.10), we find that Equation (4.7.12) is now satisfied, which completes the proof. 4.7.3. General Solutions
Finally, we demonstrate how to obtain the general solution of the nonconservative canonical system (4.6.4) from a complete solution of the basic field equation, (4.7.1). Suppose that one of the constants in the complete solution, say C1, can be considered to be an arbitrary function of all other constants
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
197
CZ,.-.,C 2 n ; C1
= Cl(C2, ..., C 2 n ) .
(4.7.13)
In what follows, we will show that a complete solution of the basic field equation (4.7.1), together with the 2n - 1 algebraic equations
au au acA ac,
-+ -DA
=
( A = 2, 3,
0
..., 2n),
(4.7.14)
defines a family of general solutions of the nonconservative canonical system (4.6.4), where
ac, acA
( A = 2 , 3,..., n)
DA = -,
(4.7.15)
is a set of new constant parameters. To prove this, we form the time derivative of (4.7.14):
a2u a2u . a2u +acAat acAaxi2 + acAaz, 2, a2u . X I. + -2.1 a2u ac, at +-ac, ax1 ac, az,
= 0.
(4.7.16)
Since, for a complete solution of the form (4.7. l), the basic field equation (4.6.10) is an identity, we can differentiate partially with respect to CA and obtain
-(? a 2 u
a2u +- a 2 u DA + aH acAat at ac, au
ax
+ -DA) a2u acA axlac,
a2u a2u
a u + ac, au (-acA
-DA)
=
0.
(4.7.17)
Employing (4.7.14), and taking the difference between Equations (4.7.16)
198
4. Motion of Conservative and Nonconservative Dynamical Systems
and (4.7.17), we find that
a2u
a2u
a2u
(4.7.18)
Therefore, under the supposition that the following determinant is different from zero: a2u
\
a2u
det( det
a 2 u + - 0 2a 2 u +-D2 ac2 axn ac, ax'
a2u +- a 2 u D2 ... -+ac2. az. ac, az. a c 2 az2 ac, az2
j
a2u a2u ac2. ax1 +-Din a c l ax1
a2u ... -+-D2n ac2. ax'
a2u ac, axn
a2u a2u a2u a2u Din ... -i-D2n ac2, az2 ac, az2 ac2. az, ac, az,
-+-
I
(4.7.19)
it follows that the 2n - 1 canonical equations ,
aH api
= -,
pa=--
aH ax. + Q ,
( i = 1 , ..., n , a ! = 2,..., n) (4.7.20)
are satisfied for arbitrary values of the constant parameters C1, ..., Czn, D2, ...,Dzn . To show that the first equation, p l = -(aH/axl) QI, is satisfied also, we form the time derivative of (4.7.1) and combine the resulting equation with (4.7.20):
+
Substituting this last expression into the basic field equation (4.6. lo), it follows that the first canonical equation is also satisfied for the arbitrary values of the C's and D's, which completes our proof. It follows from our analysis that Equations (4.7.1) and (4.7.14) contain 4n - 1 arbitrary constant parameters C1, ..., Czn and Dz, ..., D2n. Consequently, we have 2n - 1 superfluous constants remaining arbitrary during the process of finding the motion of the nonconservative dynamical system (4.6.4). The method for getting a general solution of (4.6.4), which is free of this redundancy, is contained in the following paragraphs.
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
199
Let us suppose that we have found a complete solution of the basic field equation (4.6.10) in the form of (4.7.1): p1 = U ( t , x l , . . . , x " , z ~. -, . , ~ n , ~
1
,
~2,).
(4.7.21)
Solving this equation with respect to CI, we find a relation of the form
.. .,x", U,2 2 , ...,Z n , C2, ..., C2n). (4.7.22) = 2 , 3 , ...,2n)enter linearly into (4.7.14), we can
C1 = '€'1(t,xl,
Since the constants DA (A solve these algebraic equations with respect to DA and then substitute (4.7.22) into the resulting expressions. The resulting equations are of the form D2 = Y2(t,XI,...,x", U,2 2 ,
...,Zn, C2, -..,C 2 n ) , (4.7.23)
D2n = Y2n(t,X I ,
...,x", U,2 2 , ...,Zn, C2, ..., C2n).
The relations (4.7.22) and (4.7.23) form a family of general solutions of the differential equations of motion (4.6.4) in which the parameters CA (A = 2,3, ..., 2n) are arbitrary. Therefore we are free to choose any particular value for these parameters in their relevant domain. Hence by putting U = p1, 2 2 = p2 , ...,Z n = Pn ; we have ~1
=
~ ( xl, t , - -. xn,
.- -
pn), (4.7.24)
D2n = Ylzn(f,xl,. . . , x " , P I ,. . . , p n ) .
The procedure described here for finding the family of general solutions of the nonconservative canonical equations (4.6.4) was introduced in Reference [13]. In order to clarify the previous discussion, we turn 'to an example. Let us consider the nonconservative dynamical system whose differential equations of motion are of the form 21 = X 2 ,
Xl(1) = a,
(see [15]). Taking as the field variable xz
=
U(t,XI), we find the basic field
200
4. Motion of Conservative and Nonconservative Dynamical Systems
equation in the form
-au + at
u2+ -u= 0. u-au - ax,
x1
t
(4.7.26)
The variables XI and t are separated by the assumption that (4.7.27) from which we find that (4.7.28) where an overdot denotes time differentiation and a prime denotes differentiation with respect to X I . This expression can be split into the following system:
Integrating, we find a complete solution of the basic field equation (4.7.26) of the form XI lnxl + A (4.7.29) x2 = t lnt + B ’ where A and B are constants. We now discuss the various possibilities of employing this complete solution in the same order as the observations given in Sections 4.7.1,4.7.2 and 4.7.3. i. As indicated in 4.7.1, we can interpret (4.7.29) as a bundle of conservation laws of the differential equations of motion (4.7.25). For example, taking in (4.7.25)’ A # 0, B = 0 and A = 0, B # 0, we find the following two independent conservation laws of the dynamical system (4.7.25) x2
-tlnt
XI
XI
-
- lnxl
x2 t
lnxl = A = const.
(for B = 0) (4.7.30)
-
In t = B = const.
(for A = 0).
Note that A and B can assume arbitrary values in these two conservation laws for (4.7.25). Due to this fact, we call the complete solution (4.7.29) a bundle of conservation laws for the dynamical system (4.7.25).
201
4.7. The Complete Solutions of the Basic Field Equation and Their Properties
ii. To find the motion that corresponds to the given initial conditions = a, x2(1) = b, we substitute these into (4.7.29) and calculate A = B(b/a) - lna. Putting this into (4.7.29), we obtain the conditioned form solution of the basic field equation, which is matched with the initial conditions xl(1)
xz = &t,
XI,
ln(xl/a) + B(b/a) t lnt + B
x1
a, b, B ) = -
(4.7.3 1)
where B is completely arbitrary. According to (4.7.6), we find the motion form (4.7.31) and also
a0
-=
aB
0.
(4.7.32)
It is easy to verify that, under the condition (In t equation gives XI = at@/@.
+ B)2 # 0 ,
the last (4.7.33)
To find the second coordinate as a function of time, we substitute (4.7.33) into (4.7.31). The parameter B disappears and we find x2 = bt(b/@-l 9 (4.7.34) which completes the solution of the initial value problem. iii. To find the general solution of (4.7.25) from the complete solution (4.7.29), we take A = A ( B ) . Hence, the equation
-axZ =o
(4.7.3 5)
aB
is equivalent to lnxl = -(lnt dA dB
+ B) - A
(4.7.3 6 )
By substituting this into (4.7.31), we find (4.7.37)
and (4.7.36) gives lnxl = D1lnt
+ D2
(D2
=B
g
-A).
(4.7.38)
The last two equations represent the general solution of the equations of motion (4.7.25).
202
4. Motion of Conservative and Nonconservative Dynamical Systems
4.8. The Single Solutions of the Basic Field Equation If we are able to find a solution of the basic field equation (4.6.10) that depends on a single constant C1, p1
=
U(t,xl,
-.. x", 2 2 ...,Z n , cl),
(4.8.1)
then it is clear that it represents a conservation law of the dynamical system (4.6.4). Under the condition aU/aC1 # 0, we can invert (4.8.1) to obtain the more familiar form of the conservation law: Q(t,xl,
..., x n , p l ,...,p n )
=
~
1
.
(4.8.2)
Naturally, if we are able to find 2n single solutions of the basic field equation (4.6.10), Q(t,xl,
(i = 1, ...,2n),
...,x", p i , .. .,Pn) = Ci
(4.8.3)
then the problem of finding the solution of the dynamical system (4.6.4) is solved, provided that the set of conservation laws (4.8.3) is mutually independent. That is equivalent to the requirement that the Jacobian be different from zero: det( xl,
C1,
ae.9
C2n
...,x n , P I ,
) #O.
.-.,pn
(4.8.4)
In practical applications of the field method discussed here, we are frequently able to find all of the single conservation laws more easily than a complete solution of the basic field equation.
4.9. Illustrative Examples
In this section, as an illustration of the foregoing theory, we will discuss several dynamics problems frequently arising in practical applications.
4.9.1. The Linear Damped Oscillator Consider the motion of the linear damped oscillator, whose differential equations of motion are Xi
= xz ,
X2
= -w'x~ -
2kx2,
(4.9.1)
where w and k (w > k) are given constants. Taking as the field variable
4.9. Illustrative Examples
203
x1 = U(t,xz),the basic field equation is of the form
au - (o’U + 2 k ~ 2au _ ) - - xz = 0. at ax2
(4.9.2)
In order to obtain single solutions of the basic equation (4.9.2), we suppose that they are of the form
x1 = U(t,x2) = A X 2
+f(t),
(4.9.3)
where A is an unknown constant and f ( t ) is a function of time (see [16]). Substituting these into (4.9.2) and equating the free terms and the terms containing xz to zero, we obtain, respectively,
f- d A f and (02A)’
=0
+ 2k(w2A)+ 0’= 0.
(4.9.4) (4.9.5)
From the last equation we find the following two values for A : (u2A)1= - k
+ ni,
(02A)2= - k - ni,
(4.9.6)
where n = (m2 - k2)’/’,
j =
(4.9.7)
By integrating (4.9.4), we find the following two single solutions of the basic field equation (4.9.2):
which represent, at the same time, two independent conservation laws of the dynamical system (4.9.1). It is interesting to note that the product C1C2 will yield the quadratic conservation law of energy type, considered previously (see Section 3.8): (4.9.9) (xi + 2kx1x2 + cu2x?)ezkf= const.
To obtain a completesolution of the basic field equation (4.9.1), we seek one with two unknown functions of time: XI =
U ( t , ~ z=) Fi(t)xz + Fz(t).
(4.9.10)
Inserting this into (4.9.2), we arrive at the following system: F1
-
w 2 F f - 2kF1 - 1
=
0,
F z - CO’F~FZ= 0.
(4.9.11)
204
4. Motion of Conservative and Noneonservative Dynamical Systems
The solution of this system is found to be Fl = [-k
+ n tan(nt + D l )01l ~ , (4.9.12)
Fz
=
D2e-k'
cos(nt
+ Dl) '
where D1 and D2 are constants. Thus, we find that a complete solution of the basic field equation (4.9.2) is
x1=
k
x2(-2
+
5)
tan(nt
DZe -kt + 01) + cos(nt + 01)
'
(4.9.13)
According to the discussion in Section 4.7, this expression represents a bundle of conservation laws of the linear damped oscillator (4.9.1). Putting, for example, D1 # 0, DZ= 0 and D1 = 0, DZ# 0, we obtain two independent conservation laws for (4.9.1):
x2
=x
k
l ( - z
+
5)
(4.9.14)
Dze-k' tan(nt) + cos(nt)
or, in inverted form,
-nt
(Di = O),
~
+ arctan
(2+ 3 -
-
=
D1, (4.9.15)
Finally, let us suppose that the dynamical problem (4.9.1) is formulated as an initial value problem: xl(0) = al,
x2(0) = az.
(4.9.16)
Substituting these into the complete solution (4.9.13) and calculating DZ, we find that
4.9. Illustrative Examples
205
Inserting this expression into (4.9.13), we obtain the following conditionedform solution :
)]cos ~1 - (na2/02)sin ~1 + Ial + (k u d o 2cos(nt + Dl)
e -kt . (4.9.17)
To obtain the solution of the initial-value problem (4.9.1) and (4.9.16), we form, in accordance to the rule (4.7.6), the expression
au
-=
aD1
0,
which yields the equation of motion
1
+ k ~ sin(nt) 2 n
e-kf.
(4.9.18)
In order to obtain the second equation of motion, we can, according to the theorem cited in Section 4.7.2, choose any arbitrary value for the constant D1 in (4.9.17). For example, taking D1 = 0, this expression becomes
=
x2
(-k
+ n) tan(nt) + a1 + (ka2/02)e -kt cos(nt)
o2
(4.9.19)
However, instead of selecting any particular value for D1 in the conditionedform solution (4.9.17), we can calculate this expression along the trajectory xz = x2(t) that is given by (4.9.18). Therefore, substituting (4.9.18) into (4.9.17), the parameter D I completely disappears and, after some elementary calculations, we obtain the more familiar form of the solution:
x1 =
x2 a1, a 2 , m Y
a1 cos(nt)
IX2
=R(f)
1
+ -n1 (alk + a2) sin(nt)
e-kt
(4.9.20)
206
4. Motion of Conservative and Nonconservative Dynamical Systems
4.9.2. Projectile Motion with Linear Air Resistance
Consider the motion of a heavy particle moving in a vertical plane with linear friction depending on the velocity. If x and y denote the horizontal and vertical axes, respectively, the differential equations of motion are x
pi
3
= P1, =
-kpi ,
p2
= p2 = =
z,
(4.9.21)
-kz - g ,
where k and g are given constants. Let the initial conditions be x(0) = 0 ,
p1(0) = Uo cos a,
y(0) = 0 ,
z(0) = m s i n a ,
(4.9.22)
where uo is the initial velocity of the particle and a is the initial angle of inclination. Taking as the field variable the momentum p1 = U(t,x, y , z), we arrive at the basic field equation
au
au
au
au
(4.9.23)
-+U--+~--(kz+g)-+kU=O. at ay az ax
+
Seeking a complete solution of the form U = f l ( t ) x + fz(t)y + f3(t)z f4(t) and repeating the procedure similar to the previous example, a complete solution of (4.9.23) is found in the form P1 = Ul(t,x , y , z, c1, ..., C4) -
k C ~ x+ CZY+ [ ( C d k )+ C3ek']z + (gCz/k)t + (gC3/k)ek' +
ekt -
~1
C4
(4.9.24)
Applying the initial conditions and expressing the constant C4 in terms of C1, CZand C3, we have the conditioned momentum:
1
+ ~ ~ c o s a -( 1CI) +
(4.9.25) Let us form the algebraic equations
-a=0o ,
acl
-a -0
-
ac2
0,
arSi
- = 0.
ac3
(4.9.26)
4.9. Illustrative Examples
207
However, after we finish the partial differentiation then, according to the theorem cited in Section 4.7.2, we may choose any particular values for C1, CZand C3. Taking, for example, CI = C2 = C3 = 0 in (4.9.26), we find, respectively, 1 x = - vo cos a(l - e kf ),
k
y
+ -k1z
+ -uok- h a ,
g
= --t
k
(4.9.27)
For the same particular values of C1, CZand C3, we obtain, from (4.9.25), p1
=
(vO
cos
(4.9.28)
which completes the calculation of the motion of the system. 4.9.3. A Nonconservative Coupled System Consider the following nonconservative dynamical system with two degress of freedom: 2 - 3x + 2y = 0 ji
+x
- 2y
=0
x(0) = a1,
y(0) = a2,
i ( 0 ) = 61,
j(0)=
(4.9.29)
b2.
Taking as the field x = U(t,x , y , z), where y = z, we find that the basic field equation takes the form
au + z -au + ( - x + 2 ~ )au - - 3~
au
- + U-
ax
at
ay
az
+ 2y = 0.
(4.9.30)
By a procedure similar to the method used for solving the two previous problems, the complete solution of this equation is found to be
u=x
cosh t + C1 sinh t + 2Cz cosh 2t + 2C3 sinh 2t sinh t + C1 cosh t + CZsinh 2t + C3 cosh 2t
+
cosh t + C1 sinh t - C2 cosh 2t - C3 sinh 2t 2y sinh t + C1 cosh t + CZsinh 2t + C3 cosh 2t
208
4. Motion of Conservative and Nonconservative Dynamical Systems
sinh t 2C1 cosh t + sinh 2t + C3 cosh 2t '-2sinh t + C1cosh t + sinh 2t + C3 cosh 2t -
+
C2
C2
+ sinh t + C1cosh t + C2 sinh 2t + C3 cosh 2t
*
(4.9.31)
Applying the initial conditions and expressing C4 in terms of C1, CZand C3, we find the conditioned momentum in the form x = p i = Q ~ , y , ~ , a i , a z , b i , bC z ,i , C2, C3)
=x
cosh t + C1 sinh t + 2C2 cosh 2t + 2C3 sinh 2t sinh t + C1 cosh t + C2 sinh 2t + C3 cosh 2t
+
+
+
cosh t + C1 sinh t 2y sinh t + C1 cosh t +
C2 cosh C2
+ C2 sinh 2t + C3 cosh 2t + C3 cosh 2t + 2a2) + cl(bl + 2b2) + C2(-2al + 2a2) + C3(bl - b2) sinh t + C1 cosh t + C2 sinh 2t + C3 cosh 2t
-2 sinh t - 2C1 cosh t sinh t + C1 cosh t + -(a1
2t - C3 sinh 2t sinh 2t + C3 cosh 2t
C2 sinh 2t
(4.9.32)
It is easy to verify that for C1 =
-a=uo
C2
= C3 = 0, the algebraic equations
(i = 1,2, 3 )
aCi
(4.9.33)
are equivalent to the following three equations of motion:
x
+ 2y - (a1 + 2a2) cosh t - (bl + 2b2) sinh t = 0;
x(2 sinh t cosh 2t - cosh t sinh 2t) -
+ cosh t sinh 2t) + 32 sinh t sinh 2t + 2a2) sinh t + (al + 2a2) sinh 2t = 0;
2y(sinh t cosh 2t
+ (-2al
x(2 sinh t sinh 2t - cosh t cosh 2t)
+ cosh t cosh 2t) + 32 sinh t cosh 2t + (bl - b2) sinh t + (a1 + 2a2) sinh 2t = 0. -
2y(sinh t sinh 2t
Finally, the last equation is available from (4.9.32) for C1 = CZ = C3 = 0: x = p1 = x coth t
+ 2y coth t + (al + 2az)/sinh t.
4.10. Applications of the Complete Solutions of the Basic Field Equation
209
4.10. Applications of the Complete Solutions of the Basic Field Equation to Two-Point Boundary-Value Problems
In this section we shall briefly demonstrate that the foregoing theory can be easily adapted to dynamical problems that are formulated as two-point boundary-value problems. For the sake of simplicity, we will confine our attention to problems with one degree of freedom and with differential equations of motion
x. = -aH ,
P
aP
aH
= --
ax
+ Q(t,x,p)
(0 It
IT),
(4.10.1)
together with the boundary conditions (for t = 0), (for t = T).
fi(t,x,p) = 0
f&, x , p ) = 0
(4.10.2)
If we know a complete solution (4.10.3)
P = W t ,x, c1,C2)
of the corresponding basic field equation
au a w a u aH + -+ - - Q(t,x, U)= 0 , at ax au ax
-
(4.10.4)
then by applying (4.10.3) at the initial and terminal times t = 0 and t = T and eliminating the constant CZ, we arrive at an equation of the form W[X(O),p(O), 0, Cil = %[x(T),P ( T ) ,T, Cil,
(4.10.5)
where Y1 and YZ are the solutions for CZwhen (4.10.3) is substituted into (4.10.2). The condition that this relation be satisfied for arbitrary values of C1 will lead to the missing algebraic relation that, together with the equations (4.10.2), form a complete system of algebraic equations for finding x(O), p(O), x ( T ) and p ( T ) . That is to say, the two-point boundary-value problem is transformed into an initial-value problem. This method is illustrated with the following examples. 4.10.1. Stefan-Boltzmann Thermal Radiation Problem The steady-state Stefan-Boltzmann thermal radiation problem is defined as a two-point boundary-value problem (see [17]):
x-x=o x(o) = 1,
(OSt$l), i(i)
+ x4(1)
=
0.
(4.10.6)
210
4. Motion of Conservative and Nonconservative Dynamical Systems
Taking as a field x = p = U(t, x), the basic field equation is of the form
au
au
-++--x=o. at ax
(4.10.7)
The complete solution of (4.10.7) is
+ Cl) + sinh(tc2+ Cl)
(4.10.8)
+ Cl) - xcosh(t + Cl).
(4.10.9)
p = xcoth(t
or C2 = p sinh(t
For the initial and terminal values of t, we find that (4.10.5) becomes p(0) sinh CI - x(0) cosh C1 = p(1) sinh(1 + Cl) - x(1) cosh(1 + Cl). (4.10.10)
Since this relation should be valid for arbitrary values of C1,it follows that p(0) = p ( 1) cosh 1 - x( 1) sinh 1,
x(0) = -p(l) sinh 1 + x(1) cosh 1
(4.10.1 1)
by using well-known properties of the hyperbolic sine and cosine. These two relations together with the given boundary conditions x(0) = 1,
p(i)
+ x4(1) = o
(4.10.12)
form a closed system for obtaining the missing initial values x(O), p(0) or x(l), p(1). Combining (4.10.12) with the second equation of the system (4.10.11), we find that 1 - 0. (4.10.13) x4(l) + x(1) coth 1 - -sinh 1 A root of this equation is found to be x(1) = a - 0.5686. Consequently, ~ ( 1 =) p = -a4 = -0.1045. Repeating the same process as previously described, for the initial-value problem x(1) = a,p(1) = p, we find the solution of the problem to be x = 0.5686 cosh(t - 1) - 0.1045 sinh(t - 1). 4.10.2. Diffusion in a Tubular Reactor
Consider the material balance for a tubular reactor with axial diffusion. The mathematical model is formulated as a two-point boundary-value
4.10. Applications of the Complete Solutions of the Basic Field Equation
21 1
problem (see [18], p. 322): x - 2AX - w2x = 0
1 x(0) - -*O) 212
(0 5 t 5 l),
(4.10.14)
X(1) = 0,
= 0,
where x is the dimensionless concentration, t is the axial coordinate, and A and w are given constants. Denoting p = x = U(t, x), the basic field equation is
au
-au + at
u--212u-w2x=o.
(4.10.15)
ax
The complete solution of this equation is given by p =U
=
[A
CZ ext + n coth(nt + Ci)]x + sinh(nt + Cl)’
(4.10.16)
where n = (A2 + w2)’”. Since C2 is a constant, we can calculate it from (4.10.16) and apply the result at t = 0 and t = 1. Hence, C2 =
1
:[
[ b ( O ) - A] sinh C1 - n -p(0) + 1 cosh C1
= -x(l)e-’[(A
cosh n
+ n sinh n) sinh C1
+ (A sinh n + n cosh n) cosh Cl]. Equating terms with sinh C1 and cosh C1, we obtain ip(0)
n
--p(0) 2A
+ e-’(12
cosh n
+ n sinh n)x(l) = 12
+ e-’(Asinhn + ncoshn)x(l) = n.
(4.10.17)
Therefore, by solving this system, we find the missing boundary values : x(1) =
2~ne’
(n2 + A2) sinh n
+ 2An cosh n ’
2A(A2 - n2)sinh n ’(O) = (n2 + A?) sinh n + 2An cosh n
= X(0).
Finally, from the boundary condition x(0) = X(0)/2A, we can find x(0) and the problem is reduced to an initial-value problem.
212
4. Motion of Conservative and Nonconservative Dynamical Systems
4.10.3. An Eigenvalue Problem
Consider the following eigenvalue problem (see [ 191):
2
1 + -kx t4
x(a) = x(b) = 0
= 0,
(a d t 5 b).
(4.10.18)
We again take x = U(t, x), and the basic partial differential equation is
au
au
kx
-+u-+-=o, ax
at
(4.10.19)
t4
whose complete solution is found to be (see [13]):
.
c 2
(4.10.20)
Applying the boundary conditions, we find that
+ U(b, O)b sni)(:
- U(a, O)a sin($)
=
0.
This homogeneous set of linear algebraic equations has a nontrivial solution under the condition
fi
acos-
a
-bcos-
b
4z b sin -
I&
-a sin a
=
0,
b
which gives sin($
4z
-
$)
=
0.
From this we find all possible values for k: (4.10.21)
for n an integer. To find the corresponding eigenfunctions, we first apply one of the given boundary conditions, say x(a) = 0, to Equation (4.10.20).
213
4.11. Potential Method of Arzhanik’h for Nonconservative Dynamical Systems
Calculating CZand inserting it into (4.10.20), we find that
U(a,O)acos[(&/a)
+C~I
It is easy to show that the equation aU/aCl = 0 is equivalent to
U(a,O)a fi(t x=t sin( at a ) )
fi
This expression, along with (4.10.21), produces the eigenfunctions of the problem; they have the form
x,, = Dnt sin
1
1 L
nnb(1 - a / t ) b-a
(Dn= const.).
J
Together with the eigenvalues (4.10.21), these form the solution of the problem. 4.11. The Potential Method of Arzhanik’h for Nonconservative Dynamical Systems
In this section we will present a field method suitable for the study of a nonconservative dynamical system that was introduced by Arzhanik’h [20] and called the potential method by him. Arzhanik’h’s approach is based on the fact that we are always able to convert an arbitrary system of first-order ordinary differential equations, xi = P ( t , x l , ...,x“)
(i = 1,
...,n),
(4.1 1.l)
into Hamilton’s form by introducing the redundant momenta pi and Hamilton’s function H . To do this we use the relation
H = piFi(t,x l , ...,xn),
(4.1 1.2)
where the summation convention with respect to repeated indices is assumed. It is obvious that the first group of Hamilton’s canonical equations, (4.1 1.3) are identical with (4.11.1).* The second group of canonical equations is
pi = -Pj
aFj 3
(i,j = 1,
..., n).
(4.11.4)
‘According to P. Appell[21], this form of Hamilton’s description was introduced by Liouville.
214
4. Motion of Conservative and Nonconservative Dynamical Systems
Without any change in (4.11. l), we can also introduce the Hamiltonian function. It takes the form
N
.. ., x") + f ( t , X I , ..., x"),
= piFi(t,X I ,
where f is an arbitrary gauge function depending on time t and the state coordinates X I , ...,x". We choose the momentum vector pi (i = 1, ...,n) to be a gradient (covariant) vector of a scalar function s = S(t, X I , ...,x"):
as
Pi = 2.
(4.11.5)
Then the Hamilton-Jacobi partial differential equation, (4.11.6)
for the Hamiltonian function given by (4.11.2) is of the form
as + 7 as F'(t, . xl, ...,x") = 0. at ax
(4.11.7)
-
Note that this partial differential equation simply states that the scalar function S = S(t, X I , .. .,x") is a conservation law of the dynamical system whose differential equations of motion are given by (4.11.1). According to Arzhanik'h's method, we transform the partial differential equation (4.11.7) into a new field equation that is obtained in the following way. We introduce the new field function Y ( t ,P I , ... , a n ) by means of the Legendre transformation of the function S(t, X I , ...,x") (as discussed in Section 4.2): S(t, x ' ,
...,x")
= pixi - Y ( t ,P I ,
- -. pn),
(4.11.8)
where p i = as s ,
i
X = -
av api'
as -- --ay at
at
*
(4.11.9)
Therefore, the partial differential equation (4.11.7) becomes (4.1 1 .lo)
This nonlinear partial differential equation plays a central role in Arzhanik'h's theory.
4.11. Potential Method of Arzhanik’h for Nonconservative Dynamical Systems
215
Let us suppose that a complete solution of the partial differential equation (4.11.10) is known, say,
where the Ci are arbitrary constants. A complete solution of (4.11.10) should have n + 1 arbitrary constants C i , however, one is of an additive nature, since only the partial derivatives of Y appear in (4.11.10). We will now demonstrate that a theorem similar to the Jacobi theorem given by (4.3.16) is valid in this case. That is to say, the general solution of the “canonical” differential equations (4.11.1) and (4.11.4) is given by (4.1 1.12) where Bi (i = 1, ..., n) are a new set of arbitrary parameters. As in the classical Hamilton-Jacobi method, we require that the following condition be satisfied:
The proof is simple. Totally differentiating the second relation in (4.11.12) with respect to time, we find that
a2y p , = 0. +i at aCi ap,
a2\y X
(4.1 1.14)
Since the expression (4.11.10) is an identity for a complete solution of the form (4.11.1 l), we can take the partial derivatives of (4.11.10) with respect to the Ci: a2y --
at aci
aFk a2y axs a p , a c i *
- Pk-
(4.11.15)
Combining the last two relations, we have a2y aCi ap,
-(bs
+ pk$)
=0
( i , k , s = 1, ..., n).
(4.11.16)
Interpreting this system as a system of linear algebraic equations for p , pk(aFk/ax3, we conclude that under the condition (4.11.13), the system of differential equations (4.11.4) will be satisfied. Similarly, to show that the dynamical equations (4.11.1) are satisfied, we first find the total
+
216
4. Motion of Conservative and Nonconservative Dynamical Systems
time derivative of the first relation in (4.11.12)and employ (4.11.15): . X I
a2y =api at
+-apia2y ap,
P S
(4.11.17) On the other hand, differentiating the identity (4.11.10) partially with respect to p i , we obtain
(4.11.18) Combining these last two relations, we see that the differential equations (4.11.1)are also satisfied. This completes the proof. Note that in practical applications, the existence of the superfluous variables pi makes the process of finding the general solution of the dynamical system (4.11.1)rather complicated. Namely, from the first relation (4.11.1 l), we find that
In order to find the solution in explicit form as a function of t, we have to eliminate the momenta pi from the second relation in (4.11.12):
And, to obtain a general solution of (4.11.1)of the form
(4.11.21) xi = f ' ( t , a1, .. ., an), a new set of constant parameters 0 1 , ...,an are defined as functions of Ci and Bi:
ai =
ai(C1, ..., Cn, B1, ...,Bn).
(4.11.22)
In order to illustrate the foregoing theory, we discuss the following two examples, which were also considered by Arzhanik'h [20]. 4.11.1. A Nonlinear System
Consider the nonlinear dynamical system where the pi are given constants.
4.11. Potential Method of Arzhanik’h for Nonconservative Dynamical Systems
2 17
The field equation (4.11.10) is of the form
c
-ay =
n p k P k ( E 7 .
at
k=l
(4.1 1.24)
This equation admits a separation of variables:
c n
=
where each
Yk
k=l
yk(f,Pk),
(4.11.25)
(k = 1, ...,n) satisfies the equation (4.11.26)
The summation convention with respect to repeated indices is not applied. Taking the \Yk to be of the form y k
=
Ckt
-k e k ( P k ) ,
we find that Pkpke”
=
c k ,
which means that ek =
2( C k P k / P k ) ‘ I 2 .
Therefore, a complete solution of (4.11.24) is
c n
=
Ckt
k=l
+2
According to (4.1 1.12), we have xk
=ay = apk
In order to eliminate the momenta
n k= 1
(ckpk/Pk)1/2.
(--y’2.. p k ,
(4.1 1.12) :
(4.11.27)
(4.11.28)
we use the second equation in
alu
-= B k , ack
which, in this case, is equivalent to (4.11.29)
218
4. Motion of Conservative and Nonconservative Dynamics1 Systems
Substituting this into (4.11.28),we find that xk =
and, in this case,
ak
1 Pk@k
-
(4.11.30)
t)
= Bk.
4.11.2. A First Order Equation
Consider a single equation of the first order: x = x.
(4.11.31)
The corresponding field equation is
ay ay -=p-, at ap
(4.11.32)
and a complete solution is found to be Y = Ct
+ Clnp.
a P'
B
(4.11.33)
Hence, aY aP
x=-=-
a\y = -=
Therefore, C
x = - -B
e e
-t -
ac
ae',
t
+ lnp. (4.11.34)
where
a = Ce-B. 4.12. Applications of the Field Method to Nonlinear Vibration Problems*
In this section, we apply the field method described in Sections 4.6 through 4.10 to nonlinear vibration problems, combining the method described here with the well-known multiple-scale asymptotic method (see [lo], [22]). For the sake of simplicity, we confine our considerations to autonomous dynamical systems with a single degree of freedom, whose canonical *The material discussed in this and the next section can be found in References [23] through ~51.
4.12. Applications of the Field Method to Nonlinear Vibration Problems
219
differential equations of motion are of the form X
=p,
= -O;X
+ Ee(x,p),
(4.12.1)
where p is the momentum, x is the generalized coordinate, E is a small constant parameter that represents a measure of the relative importance of the nonlinearity, and 6’ is a given function depending on x and p . Since both variables play an equal role, we have two possibilities for choosing the field variables. We shall consider, in some detail, each of the two possible cases. 4.12.1. The Field Momentum Approach
First, we suppose that the momentum p can be represented as a field depending on time t and the generalized coordinate x : (4. 2.2)
p = X = a,(t,x).
For this case, the basic partial differential equation is of the form aa,
-
at
aa, + a+ O ~ X= Ee(x,p). ax
(4. 2.3)
It is assumed that we are not immediately able to find an exact complete solution of this field equation. Therefore, we will try to establish a method for finding an approximate asymptotic solution by taking advantage of the fact that the term on the right-hand side of (4.12.3) is small. Let us assume the following asymptotic expansions for @ and x in terms of the small parameter E :
+ &@,(XI,T , r) + E’@Z(XZ,T , r) + x(t, E ) = xo(T, r) + Ex,(T, r) + E’xz(T, r) + ...
iP(t,x, E ) = OO(XO,T , r)
(4.12.4) (4.12.5)
where T and r are the “fast” and “slow” time scales, defined T = t(l
+ O~E’ + 0
3
+~
* * a ) ,
~
(4.12.6) (4.12.7)
t = Et.
The quantities CUZ, 0 3 , ... are unknown constant parameters. The general procedure to be followed can be considerably simplified if we impose the following requirement on the linear and unique dependence of the field momentum components Oi of X i ( i = 1,2, ...,n): aa, - aa0- aa,, ax ax0 ax,
amn ... = axn
(4.12.8)
220
4. Motion of Conservative and Nonconservative Dynamical Systems
The time derivative is transformed according to
a
- = (1
at
+02E +W E 3 + 2
a**)-
a + E- a aT at'
(4.12.9)
In order to assure the validity of (4.12.1), we find the total time derivative of (4.12.5) and equate it to (4.12.4). By equating like powers of E , we obtain the following compatibility conditions:
axo aT
@O(XO , T, T ) = -,
(4.12.1 0) (4.12.11)
ax2 axo axl @2(x2,T, t ) = -(T,t )+ 0 2 -(T,7) + -(T, TI, aT aT aT
(4.12.12)
and so on. Since, according to (4.12.2), we have nominated the momentum as a field function, it is natural to suppose that the quantities axl/aT, ..., ax,,/aT can be interpreted as new field functions depending on T, t and the corresponding X i . That is to say,
axl = @ ? ( X i , r, t ) , aT (4.12.13)
ax, _ - a,*(~,, T, TI. aT
Therefore, the compatibility conditions (4.12.10) through (4.12.12) become (4.12.14)
axo (92(x2,T, t )= M ( x 2 , T, t)+ 02-(T, aT
t)
axl + -(T, at
t ) , (4.12.16)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
221
From the last two equations, we have, by partial differentiation,
(4.12.17)
But on the basis of (4.12.8), we also have (4.12.18)
Using Equations (4.12.14) through (4.12.16) and (4.12.9), substituting (4.12.4) and (4.12.5) into the basic field equation (4.12.3), and then equating coefficients of like powers of E , we obtain the following chain system of partial differential equations:
ao0+ 0 0 -d o o + 002x0 = 0; aT axo
a@;
aT
(4.12.19)
azxo +--) axo ao0 (=+= aT axo
a@., a@; + 0;- ax2 + 002x2 = -0 ax,
at
-
a
axo
azxl aTat
(4.12.20)
a@? a2x0 a t at2
&(T, t,XO,@o, XI, @?I;
(4.12.21)
where 81, 82, ... denote the expansions of the function 8 into powers of E . The right-hand sides of these systems are known functions of both T and t at each stage of the development. The system of the partial differential equations (4.12.19) through (4.12.21) can be solved in the sense that, for the each equation of the system, we can find a complete solution. After adjusting it to the given initial conditions, we apply the rule (4.7.7), which produces a corresponding correction term for x. Obtaining xi = Xi(T, t)in this way and substituting it into the corresponding complete solution, that is, calculating the momentum @'! along the trajectory, we also obtain the related correction term for the momentum. Putting this into the next equation of the system, we repeat the calculations.
222
4. Motion of Conservative and Nonconservative Dynamical Systems
Because of the enormous increase in the algebraic calculations in each new step, we suppose, as is generally accepted in perturbation theory, that only a couple of the correction terms in the expansions (4.12.4) and (4.12.5) are necessary for a reasonably good approximate solution to the problem. In order to find a complete solution of Equation (4.12.19), we assume that it takes the form @O(XO, T, T ) = xofo(T) + FoV, t ) ,
(4.12.22)
where f o ( T ) and Fo(T, t ) are unknown functions, Inserting this into (4.12.19) and equating the terms containing xo and the free terms to zero, we arrive at the following system: dfo dT
-
+ f i + 00'= 0,
aF0
a T + fOF0
= 0.
(4.12.23)
COS(UOT + C)'
(4.12.24)
Integrating, it follows that f o = -00
tan(o0T + C ) ,
Fo =
where C is a constant and D is a function of the slow time t. Therefore, we have a complete solution of (4.12.19) in the form Po
= @O =
-x000 tan(ooT + C ) +
D(t) COS(COOT + C)
(4.12.25)
Let the initial conditions applied to the system (4.12.1) be 4 0 , E ) = a,
or
P(0, &) = B,
xo(0) = a,
Xi(0) =
PO(O) = P,
Pi(())= 0,
0,
2.26)
2.27)
for i = 1,2,3, .... Since we are interested in the conditioned form of the solution of the basic equation (see Section 4.7.2), we insert (4.12.27) into (4.12.25) and obtain D(0) = p cos C + a 0 0 sin C , (4.12.28) where we have used the obvious relation @0(0,0, a) = p, which follows from the nomination of the field variable p = @ ( t ,x). The form of Equation (4.12.8) indicates the structure of the slow function D(t), which is selected in the spirit of the variation of the arbitrary constants, D(t) = B ( t ) cos C + A ( @sin C . (4.12.29)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
Comparing the last two equations for T B(0) = p,
=
223
0, we find, by initial conditions,
A(0) = a o o .
(4.12.30)
It is to be noted that the guiding principle that determines A(7) and B(r) is that @ I is to be free of secular terms of the type T sin o o T and T cos OOT, because they lead to unboundedness with respect to T. The differential equations thus obtained should be integrated subject to the initial conditions (4.12.31). Using (4.12.29) for D(s), the conditioned form of the momentum (4.12.25) becomes @O(XO, T, 7) =
-xooo tan Y
where
+ B(r)cos Ccos+YA(r)sin C , (4.12.31)
+
Y = COOT C .
(4.12.32)
According to the rule (4.7.6), we can easily verify that the equation
a@o - -0 ac
(4.12.33)
will lead to the equation 00x0
= A(7)cos o o T
+ B(t)sin o o T
(4.12.34)
In fact, taking into account the following identities: cos c COS(CUOT + C ) = cos o o T + sin o o T tan(o0T + C ) sin C COS(COOT + C ) = -sin o o T + cos ooT tan(ooT + C ) , we can write (4.12.31) as
40(x0, T, r) = tan(w0T + C)[-XOOO + A(7)cos COOT+ B(t)sin COOT]
+ B(T)cos OOT,
- A ( t ) sin COOT
and the equation (4.12.34) is immediately verified. At the same time, inserting (4.12.34) into the last equation [or in (4.12.31)], we find the momentum along the trajectory: po(T, r) = 4oIx0= - A ( t ) sin o o T + B(T)cos COOT. (4.12.35)
224
4. Motion of Conservative and Nonconservative Dynamical Systems
Substituting (4.12.31) and (4.12.34) into the right-hand side of (4.12.20), we obtain the basic field equation in the form
a+?
+ 4?+ oixl aT ax1
=K
(
dA dB A , B, - - T, 7 , C ) d7’ d7’
(4.12.36)
where K(A, B, d A / d t , dB/dr, T, 7 , C ) denotes the group of terms (4.12.37)
In order to find a complete solution of (4.12.36), we put
4:
= fo(T)xi
+ K ( T ,7),
(4.12.38)
where, according to the requirement (4.12.18), the function fo(T) is given by the first relation in (4.12.24) and Fl(T, 7 ) is an unknown function of both T and 7 . By substituting (4.12.38) into (4.12.36), we obtain
am
-=
aT
dA dB K ( A , B, -, -, T, t, d7 d7
+gcosooT+
(4.12.39)
where (4.12.40)
As mentioned previously, the functions A(7) and B(7) may be obtained by the requirement that no secular terms appear in the differential equations. Integrating (4.12.39) and repeating the same process as in the previous step, we find @ ? ( X I , T, t, C )and match it with the initial conditions. Applying the rule aO?/aC = 0, we obtain x1= xl(T, 7 ) . Next we calculate the momentum and substitute these quantities into the rightalong the trajectory, @? hand side of (4.12.21). The same process of calculations is then repeated.
Ix1,
THEVAN DER POLOSCILLATOR As an illustration of the previous theory, we turn to van der Pol’s nonlinear and nonconservative oscillator : x =p,
p
=
-x
+ E(1 - x Z )p .
(4.12.41)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
225
Taking the field momentum approach, we put p = @(t,x), the basic partial differential equation is then
a@
a@ + x
- + 0at ax
= E(1 - x2)@(t,x).
In this case, we have 00
0 = (1 - X2)@(t,x),
= 1,
Y =T
(4.12.42)
+ C;
(4.12.43)
therefore, using (4.12.3) and (4.12.4), we find that
el = (1 -
(4.12.44) (4.12.45)
From (4.12.31), we have
+ dA/dtsin C cos Y
a@,- - dB/dtcos C at
dB dA =-cosT--sinT+ dr dr
tanY
.
(4.12.46)
In evaluating this expression, we have employed the identities given after Equation (4.12.34). The quantity K defined by Equation (4.12.37) now becomes
dA dr
= 2-sin
T-
- [l - (A cos T + B sin T)’]( -A sin T + B cos T),
(4.12.47)
and Equation (4.12.38) becomes dA dt
aMl = k-sin aT
-
dB T - 2-cos dr
T
I
[I - (A cos T -k B sin T ) 2 ] ( - Asin T + B cos T) cos Y
[ (2
A3 1 = sinT 2--A+-+-AB2 4 4
1
+ 4-1 cos 3T(B3 - 3d2B)
- 3AB2) cos Y .
(4.12.48)
226
4. Motion of Conservative and Nonconservative Dynamical Systems
Had we integrated this equation with respect to T, considering the functions A ( t ) and B(7) as constants, the terms containing sin T c o s Y and cos T cos Y (with Y = T + C ) would have produced secular terms of the form T sin T and T cos T. In order to avoid this, we choose A 2 + B 2) = O ,
2 dA --A(ldt
A 2 + B 2) = O .
- 2 -dB +B(ldt
(4.12.49) Multiplying the first equation by A and the second by B and subtracting, we find that
);
d dt
-(d)-
-
=
+ B2(t).
0,
a2 = A 2 ( t )
(4.12.50)
Integrating this equation, it follows that a2 is of the form 4
2
a =
1
+ [(4/a02) - 1 1 e - ~ ’
a? = A(0)2+ B(0)2. (4.12.51)
Now we introduce the so-called van der Pol variables, A ( t ) = a(t)cos a,(?) and B ( t ) = a(r) sin a,(t), where a(t) is the amplitude and p(t) denotes the phase angle. Substituting these into (4.12.49), we obtain da,/dt = 0, or a, = a , ~= const. Therefore, Equations (4.12.31), (4.12.34) and (4.12.35) become, respectively,
00= -xotanY
C* + a-sin COSY ’
(4.12.52)
xo =acos T * , PO
=%Ixo
(4.12.53)
= -asin T * ,
(4.12.54)
with C* = C - 90,
T*
=
T+
~ 0 ,
Y = T + C = T*
+ C*.
(4.12.55)
Equation (4.12.48) becomes aMl - -
aT
*a3 sin 3T* cos Y = *a3 sin 3T* cos(T*
+ C*).
(4.12.56)
Integrating with respect to T * , taking into account that dT = dT* and
227
4.12. Applications of the Field Method to Nonlinear Vibration Problems
keeping in mind that 0;= fox1 @?(XI,
T, r) =
-XI
+ (Ml/cos Y ) , we find that
1 tan Y - - a 16
cos(2T* - C*) + cos(4T* cos Y
+ C*) (4.12.57)
Observing (4.12.53), we can write (4.12.15) in the form Ql(x1,
da
T, 5) = @?(.XI,T, r) + -cos dr
T*.
(4.12.58)
Hence, since xl(0) = Q1(O) = 0, we find da Qf(0) = - -(0) cos @I, dt
and from (4.12.57) for t = 0 (i.e., T = 0, D~(o)=
da -z (0)cos
90
cos C
T =
0), we have
+ & ~ ( o ) ~ [ c o -s (C~ )~+ 9 COS(3po + c)]. (4.12.59)
As in the previous step, we assume that the function DI(T)is equal to D I ( T )= BI(T)cos C*
+ AI(T)sin C*,
(4.12.60)
where Bl(r) and AI(T)are unknown functions of the slow time, to be determined in the next step of the calculation. Putting r = 0 in (4.12.60) and then comparing the terms containing sin C and cos C with those in Equation (4.12.59), we can easily find Al(0) and Bl(0). Combining (4.12.60) and (4.12.57), we have 1 Ql(x1, T, T ) = -XI tan Y - - a 16
C O S ( ~ T-* C*) + ~ c o s ( ~ T+ *C*) cos Y
) C* + B1(r) cos C*cos+YA I ( Tsin
(4.12.61)
It is easy to verify that the equation aQl/aC* = 0 is equivalent to XI
= AI(T)cos T*
a3 + Bl(r) sin T * - -sin 32
3T*.
(4.12.62)
228
4. Motion of Conservative and Nonconservative Dynamical Systems
Substituting (4.12.62) into (4.12.61), we find the first correction term for the momentum along the trajectory:
@?Ix,
= Bi(t) cos T* - Al(7) sin T* - &sin 3T*.
(4.12.63)
Note that in calculating (4.12.62) and (4.12.63), the identities given after Equation (4.12.34) can be of considerable value. To find a complete solution of the basic field equation (4.12.21), we assume that the solution is of the form @2
= fo(T)x2
+ F2(T, t) = -x2
+ M2(T’ ‘I. cos Y
tan Y
(4.12.64)
The partial differential equation (4.12.21) now becomes
a~~
1 -COSY a T
aTat
ar
at2
(4.12.65)
Substituting (4.12.52), (4.12.53), (4.12.57) and (4.12.63) into this equation, it follows that *=[(2ao,-2--aT
dB1 dt
- - a 3 ,da 1 a 2B l + B l + - da 4 dt 4 dt
3 5 + -a2A1sin3T* + -a5cos5T* 4 128
I
COSY.
-
$a2)
(4.12.66)
Elimination of the secular terms necessitates satisfying the following equations: d2a dBi 2- Bl(1 - *a’) = 2ao2 - 7(1 dt dt
dA 1 2- Al(1 - $a2) = 0. dt
+ &a5;
(4.12.67) (4.12.68)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
229
By means of (4.12.50), we can express the last two equations as follows:
dB1 2 da 2--.---B d7 a d7
1
-
dA 1 d7
2a(wz
(:$
da + &-) - (&a2 - $)--. d7'
- l>,,
= 0.
(4.12.69) (4.12.70)
In order to avoid the appearance of secular terms, we choose 0 2
(4.12.71)
=16 1.
Now, integrating (4.12.69) and (4.12.70), we find
+ + a h a - &a3,
Bl(7) = -bla
Al(7) = ala3e-',
(4.12.72)
where a1 and bl are constants of integration. The first correction terms, (4.12.62) and (4.12.63), now become XI =
a1a3e-'cos T*
O I =~-&cos3T* ~ ~
+ ( i a l n a - bla - &a3) sin T* - &a3sin 3T*;
(4.12.73)
+ ( i a l n a - bla - &a3)cos T* - ala3e-'sin
T*. (4.12.74)
Therefore, for a second approximation,
x = acos T* + &[a1a3e-'cos T* - &sin3T*] + 0 ( e 2 ) ,
= -asinT* -
+E
[
&a3 sin 3T*
-&cos3T*
da + -cos d7
T*=t(1-;),
7=&t,
(4.12.75)
+ ( i a l n a - bla - &a3)sinT*
1+
T*
where
+ ( i a l n a - bla - &a3)cos T*
0(c2),
a=
(1 +
(4.12.76)
2 [(4/ao2) - l ~ e - ' ) " ~ ' (4.12.77)
This approximate solution is in full agreement with the results obtained in Nayfeh's monograph [l 11 by means of the method of multiple scales.
230
4. Motion of Conservative and Nonconservative Dyoamical Systems
4.12.2. The Field Coordinate Approach
As a second possibility, consider the case in which the coordinate x plays the role of the field variable; x = U(t,p). (4.12.78)
The basic field equation corresponding to the dynamical system (4.12.1) now takes the form
au + au [O;U+ EB(U,p)] - p
-
at
ap
(4.12.79)
= 0.
Let us assume the following asymptotic expansions for small values of the parameter E : U(t,p,E )
=
Uo(po, T, T) + EUi(p1, T, 7) + E ~ U ~ (T,PE )~+,
p(t, E ) = po(T, 7 )
+ EPI(T,7 ) + c2p2(T,4 + .-.,
*.*,
(4.12.80) (4.12.81)
where T and 7 the “fast” and “slow” times defined by (4.12.6) and (4.12.7). As with the other field variable, we simplify the analysis by assuming that (4.12.82)
Substituting (4.12.80) through (4.12.81) into (4.12.79) and equating like powers of E , we obtain the following system of basic field equations:
avo
aT
auo
(4.12.83)
OoUo-Po = 0; apo
auo- 91 -, avo. aT at aP0 au2 woU2-au2- p2 = -02- auo- aul - 92-; auo --
aul
--
aT
aul
OoU1-p1 = aPl
ap2
--
aT
a7
apo
(4.12.84) (4.12.85)
where 0 1 , 1 9 2 , ... denote the terms of the expansion of the function 9. It is clear that this system of partial differential equations has a considerably simpler structure than the system (4.12.19) through (4.12.21), due to the absence of the compatibility conditions (4.12.14) through (4.12.17). The general plan for obtaining an approximate solution remains unchanged, and it is described by the field momentum approach. For example, the
4.12. Applications of the Field Method to Nonlinear Vibration Problems
231
complete solution of the basic field equation (4.12.83) is found to be Po Uo(p0, T, t) = -tan 0 0
Y
where C is a constant and Y = woT B ( t ) satisfy the initial conditions
+ A ( @cos Ccos-YB(t)sin C ,
(4.12.86)
+ C . The unknown functions A ( t ) and Po = -. P B(0) = -
A(0) = a = xo(O),
(4.12.87)
wo
0 0
In order to illustrate this technique, we consider the following example. OSCILLATOR WITHSMALLCUBICDAMPING
Let us consider an oscillator with small cubic damping, whose differential equation of motion and initial conditions are given by References [lo] and [25]: x =p,
x(0) = 1 ,
P = - x - &P3
9
(4.12.88)
P(0) = 0.
With the field variable x = U(t,p),the basic field equation is
au - - u-au - p at
aP
au
= cp3--.
(4.12.89)
aP
To lowest-order terms, the system of partial differential equations (4.12.83) through (4.12.85) for this case becomes (4.12.90)
a ul
--
aT
a
-u2 -
aT
u1-a ul - p 1 = aPl
au2- p2 = -
u2-
aP2
avo+ po-; ,auo at apo
(4.12.91)
--
+ 3p;pi-,avo apo
(4.12.92)
where the notation (aUo/aT)I,, means that the quantity aUo/aT should be evaluated along the trajectory PO = po(T, t),which is obtained from the previous step. The conditioned-form solution is given by (4.12.86) for 00 = 1 . From the same equation, applying the rule aUo/aC = 0 and calculating the field
232
4. Motion of Conservative and Nonconservative Dynamical Systems
variable UOalong the trajectory, we find that
PO = - A ( t ) sin T + B ( t ) cos T,
(4.12.93)
xo = UoL, = A ( r ) cos T
(4.12.94)
+ B(r) sin T,
with A(0) = 1 ,
B(0) = 0 .
(4.12.95)
Substituting (4.12.86)and (4.12.95)into the right-hand side of (4.12.91) and looking for a complete solution of the form UI = PI tan Y
7) + Ml(T, COSY
(Y = T
+ C),
(4.12.96)
Equation (4.12.91) is reduced to aM1 = -dA cos C
aT
dt
=
dB + -sin C + [ - A sin T + Bcos TI3sin Y dr
(2
cos C --
- A 3sin4T + 3A2B sin3Tcos T - 3AB2 sin T cos3T
)
+ B3 sin T cos3T + sin C
(2
- - A 3 sin3Tcos
T
+ 3A2Bsin2Tcos2T - 3ABZsin T cos3T + B3 Integrating this equation with respect to T and considering the slow time as a constant, we find that Ml(T,x) = c o s C -
1
T-A3(-asin2T+&sin4T)
dB
sin C [ [ dt
+ i3B ( A 2
+ Bz)
t
1
(4.12.97)
where Kl(r) is an arbitrary function of the slow time (or U I )is unbounded as T + 00 unless dA dt
-
+ -38A ( A 2 + B 2 ) = 0 ,
dB 3 - + -B(A2 dr 8
+ Bz) = 0 .
t. Note
that Ml(T, t)
(4.12.98)
4.12. Applications of the Field Method to Nonlinear Vibration Problems
233
The solution of this system, subject to the initial conditions in (4.12.95), is A(?) =
2
(4
B(7) = 0.
+ 37)”2’
(4.12.99)
Thus, the expression U I ,given by (4.12.99, becomes VI(PI, T, 7) = P I tan Y - A 3
COS
c(&sin 4 T - 3sin 2T) cos Y
4T + A 3 sin C(&coscos Y
cos 2T)
K(7) +-cos Y’
(4.12.100)
where -&A3(?) sin C + Kl(7) = K(7). Since U1(0,0,0) = x1(0,0)= 0, andpl(O, 0) = 0, we find from (4.12.100) that for t = 0 (that is, T = 0, 7 = 0), we have K(0) = & sin C . As in the previous step, we suppose that the function K(r) can be expressed K(7) = AI(T)cos C - Bl(7) sin C,
(4.12.101)
with Ai(0) = 0,
Bi(0)
=
6,
(4.12.102)
where the slow variable functions A l ( r )and &(?) will be determined in the next step of the approximation. Thus, the conditioned-form solution of (4.12.91) is Ul
=pltanY -
cos C(& sin 4 T - $ sin 2T) cos Y
4 T - # cos 2T) Al(7) cos C - Bl(7) sin C + A3 sin C(& coscos + Y cos Y
(4.12.103)
It is easy to verify that the condition dUl/dC = 0 yields pl(T
7) =
&A3 cos 3 T - &A3 cos T - A1 sin T
+ B1 cos T.
(4.12.104)
Substituting this into (4.12.102), we find that xl(T,
7) =
U1 I p l
= &A3 sin 3 T
+ &A3 sin T + A1 cos T + BI sin T.
(4.12.105)
234
4. Motion of Conservative and Nonconservative Dynamical Systems
Now, from (4.12.93), (4.12.94) and (4.12.86), we have po(T, r) = -A(7) sin T, XO(T, T) =A(?)cos T,
(4.12.106)
A ( t )cos C UO(PO,T, 5 , C ) =POtan Y + cos Y
(Y = T
+ C).
Forming the partial derivative with respect to T in the third equation, having substituting in the result from the first, we obtain A(T)sin C
(4.12.107)
Looking for a complete solution of (4.12.92) of the form U2 = p ~ t a n Y+ cos Y M2(T,
T)
(4.12.108)
and substituting (4.12.103) through (4.12.107) into the right-hand side of (4.12.92), we have, by collecting only the resonant terms,
rz
27 + sinC - + -38 A ~ B--A' ~ 256 + (nonresonant terms).
-
(4.12.109)
To eliminate the secular terms, we require that dA1 dr
-
dB1 dr
-
3 27 + -A2B1 - -A2 8 256
+ -98 A -
~ A= ~ 0,
0 2 A = 0.
(4.12.110)
The solutions of these equations, subject to the initial conditions (4.12.102), are A I ( T )= 0, Bl(t) = -&A3 = -Q(4
Therefore, for t
+ S A + A02t + 3?)-3/2 + $4 + 3r)-1'2 + 202t(4 + 3t)-'l2.
(4.12.111)
-, 00, the function Bl(t) will be unbounded unless 0 2 = 0.
4.13. A Linear Oscillator with Slowly Varying Frequency
235
Thus, we have the following approximations to the second order in P
= Po
+ EPl
= - A ( t ) sin T
E:
+ &[&A3cos 3T + (-%A3 + S A ) cos TI
x = UOL, + U1Ll = xo
+ EX1
+ &[&13sin3T+ (&A3 + %A) cos TI, (4.12.112) where A(r) is given by (4.12.98) and T = t + O(e3).This result is in full = A cos T
agreement with the expansion obtained by Kevorkian and Cole (see [lo], p. 123). Concluding this section, the following two remarks are of interest. First, the asymptotic method demonstrated in this section is based on the supposition, generally accepted in all asymptotic methods, that the successive approximations are developed formally by a recursive procedure for a fixed number of terms and for E -, 0. In other words, we require that the asymptotic expansions (4.12.4) and (4.12.5) represent a reasonably accurate solution for a long interval of time, t + 00. Second, we can easily extend our considerations to nonautonomous systems. Since this extension is straightforward, we will demonstrate it by means of an example in Section 4.13. 4.13. A Linear Oscillator with Slowly Varying Frequency
As an example of a nonconservative, nonautonomous, linear dynamical system, we consider an oscillator with a slowly varying restoring force, whose differential equations are given by x
= y,
(4.13.1)
j , = - K( t) x ,
where t = Et is the slow time and E is a small parameter. We suppose that K ( t ) = 0(1) (in E ) and is strictly positive on the entire time interval. Taking as the field variable x = U(t,y ) , the basic field equation becomes
au - K(7)U-au - y = 0. _ at aY
(4.13.2)
We assume that there exists an asymptotic representation for U and y of the form
U(t,Y , E ) = uo(yo, T, t) + EUICYI, T,
~ ( tE ), = yo(T, 4 + EYI(T, +
*.*
,
+
,
(4.13.3)
236
4. Motion of Conservative and Noneonservative Dynamieal Systems
where we introduce, as suggested in Reference [ l l ] (p. 282), the fast time scale 1 T = -f(r),
(4.13.4)
&
where f(r) is an adjustable function of the slow time 5. Assuming, as before, that aU/ay = aUo/ayo = aUl/ayl = - - -we , obtain the following system of partial differential equations by equating the coefficients of the lowest powers of E to zero: (4.13.5) (4.13.6)
To find a complete solution of (4.13.5), we assume a form similar to that of the previous cases:
UO= A(7)yOtan Y
+M(r) cos Y
(Y = T + C ) ,
(4.13.7)
where A(r) and M(r) are to be determined. Substituting (4.13.7) into (4.13.5), we find that - 1
[
:- 1
+ tanZY A(?) - - K(r)A2(r)
+
(4.13.8)
This relation will be satisfied identically for all Y, yo and M(r) if
df
A(7)-
dr
- 1 =
0.
A(r)-df - K(r)A2(t)= 0, dt df-
dr
From this system, we find that
A(r)K(r)= 0.
237
4.13. A Linear Oscillator with Slowly Varying Frequency
Therefore, (4.13.9)
As discussed previously, we select the function M(r) to be
M(r) = A(r) cos C
-
B(r)sin C .
(4.13.10)
Consequently, the conditioned-form solution of the basic field equation (4.13.5) is UO =
&tan
Y
+ A ( r )cos Ccos-YB(r)sin C .
We easily verify that the condition YO
=
(4.13.11)
auo/ac= o leads to
-m[A(r)sin T - B(r)cos TI.
(4.13.12)
Inserting this into (4.13.1l), we also have xo = UoL,, = A(r)cos T
+ B(r)sin T.
(4.13.13)
Taking into account the requirement aUo/ayo = aUl/ayl, we seek a complete solution of (4.13.6) of the form (4.13.14)
Substituting (4.13.14) and (4.13.11) into (4.13.6) and collecting the secular terms, we find that
( r ) 1 dK --dA - Adr
dB B(r) 1 dK -+--dr 4 K(r) dr
4 K(r) dr
+ (nonsecular terms).
1
Thus, we have
dA dr
1 1 dK 4 K(r) dr
- + - --A(r)
= 0,
dB dr
1 1 dK 4 K(r) dr
- + - --B(r)
= 0.
(4.13.15)
Integrating, we find that (4.13.16)
where D1 and DZare constants of integration.
238
4. Motion of Conservative and Nonconservative Dynamical Systems
Hence, (4.13.12) and (4.13.13) lead to
(4.13.17)
The same result was obtained in Nayfeh’s monograph (see [ l l ] p. 282) by means of a generalized version of the multiple scale method.
References 1. Courant, R. (1962). Methods of Mathematical Physics, Vol. II, Partial Differential Equa-
tions. Interscience Publishers, New York. 2. Amol’d, V. I. (1980). Mathematical Methods of Classical Mechanics. Springer, New York. 3. Ray, J. R. (1973). “Modified Hamilton’s Principle,” American Journal of Physics 41, 1188-1 190. 4. Gelfand, I. M., and Fomin, S. V. (1963). Calculus of Variations. Prentice-Hall, Englewood Cliffs, N.J. 5. DjukiC, Dj. (1975). “A Note on Hamilton’s Principle,” Mathematica Japonicae 20(2), 132-140. 6. Bryson, A. E., Jr., and Ho, Yu-Chi. (1975). Applied Optimal Control. Hemisphere Publishing Corporation, Washington. 7. Meyer, G. H. (1973). Initial Value Methods of Boundary Value Problems. Academic Press, New York. 8. Denman, H. H., and Buch, L. H. (1973). “Solution of the Hamilton-Jacobi Equation for Certain Dissipative Classical Mechanical Systems,” J. Math. Phys. 14(3), 326-329. 9. Vujanovic, B., and Strauss, A. M. (1981). “Linear and Quadratic First Integrals of a Forced Linearly Damped Oscillator with a Single Degree of Freedom,” J. Acoust. SOC. Am. 69(4), 1213-1214. 10. Kevorkian, J., and Cole, J. D. (1980). Perturbation Methods in Applied Mathematics. Springer-Verlag, New York. 1 1 . Nayfeh, A. H. (1973). Perturbation Methods. Wiley, New York. 12. Vujanovic, B. (1981). “On the Integration of the Nonconservative Hamilton’s Dynamical Equations,” Int. J. Eng. Sci. 19(12), 1739-1747. 13. Sutela, T., and Vujanovic, B. (1982). “Motion of a Nonconservative Dynamical System Via a Complete Integral of a Partial Differential Equation,” Tensor, N.S. 38, 303-310. 14. Vujanovic, B. (1982). “Conservation Laws and a Hamilton-Jacobi-Like Method in Nonconservative Mechanics,” In Avez, A., Blaquiere, A., and Marzollo, A. Dynarnical Systems and Microphysics-Geometry and Mechanics. Academic Press, New York. 15. Ince, E. L. (1963). Integration of Ordinary Differential Equations, seventh edition. Oliver and Boyd, Edinburgh. 16. Garabedian, P. R. (1964). Partial Differential Equations. Wiley, New York.
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17. Dyer, P. (1970). The Computational Theory of Optimal Control. Academic Press, New York. 18. Finlayson, B. A. (1972). The Method of Weighted Residuals and Variational Principles. Academic Press, New York. 19. Chuev, M. A. (1975). “The Hamilton-Jacobi Method and a Problem in Solving Ordinary Differential Equations,” Differential Equations. XI(12), 2183-21 88. 20. Arzhanik’h, I. S. (1965). Field of Impulses. “Nauka” Tashkent. 21. Appell, P. (1953). Trait6 de MPcanique Rationelle, Tome Deuxihe. Gauthier-Villars, Paris. 22. Cole, J. D. (1968). Perturbation Methods in Applied Mathematics. Blaisdel, Waltham, Mass. 23. Vujanovic, B. (1984). “A Field Method and Its Application to the Theory of Vibrations,” Int. J. Non-Linear Mech. 19(4), 383-396. 24. Vujanovic, B., and Strauss, A. M. (1984). “Study of the Motion and Conservation Laws of Nonconservative Dynamical Systems Via a Field Method,” Hadronic J. 7 , 163-185. 25. Vujanovic, B., and Strauss, A. M. (1987). “Applications of a Field Method to the Theory of Vibrations,” J. of Sound and Vibration 114(2), 375-387.