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Chapter 14: Classical Mechanics
Chapter
14 Chssical Mechanics
The goal of this very short chapter is to do two things: (1) to give a statement of the famous n-body problem of celestial mechanics and (2) to give a brief introduction to Hamiltonian mechanics. We give a more abstract treatment of Hamiltonian theory than is given in physics texts; but our method exhibits invariant notions more clearly and has the virtue of passing easily to the case where the configuration space is a manifold. $1. The n-Body Problem
We give a description of the n-body “problem” of celestial mechanics; this extends the Kepler problem of Chapter 2. The basic example of this mechanical system is the solar system with the sun and planets representing the n bodies. Another example is the system consisting of the earth, moon, and sun. We are concerned here with Newtonian gravitational forces; on the other hand, the Newtonian n-body problem is the prototype of other n-body problems, with forces other than gravitational. The data, or parameters of this system, are n positive numbers representing the masses of the n bodies. We denote these numbers by ml, . . . , m.. The first goal in understanding a mechanical system is to define the configuration space, or space of generalized positions. In this case a configuration will consist precisely of the positions of each of the n bodies. We will write xi for the position of the ith body so that z i is a point in Euclidean three space (the space in which we live) denoted by E. Now E is isomorphic to Cartesian space R*but by no natural isomorphism. However E does have a natural notion of inner product and associated norm; the notions of length and perpendicular make sense in the space in which we live, while any system of coordinate axe8 is arbitrary.
288
14.
CLASSICAL MECHANICS
Thus Euclidean three space, the configuration space of one body, is a threedimensional vector space together with an inner product. The configuration space M for the n-body problem is the Cartesian product of E with itself n times; thus M = ( E ) ”and x = ( 5 1 , . . . , xn), where xi E E is the position of the ith body. Note that x, denotes a point in El not a number. One may deduce the space of states from the configuration space as the space TM of all tangent vectors to all possible curves in M . One may think of T M as the product M X M and represent a state as ( x , v ) E M X M , where x is a configuration as before and v = (vl, . . , vn), v i E E being the velocity of the ith body. A state of the system gives complete information about the system at a given moment and (at least in classical mechanics) determines the complete life history of the state. The determination of this life history goes via the ordinary differential equations of motion, Newton’s equations in this instance. Good insights into these equations can be obtained by introducing kinetic and potential energy. The kinetic energy is a function K : M X M + R on the space of states which is given by
.
~ ( 2 V ,)
=
’c 2
mi
i-1
I v i 12.
Here the norm of v ; is the Euclidean norm on E. One may also consider K to be given directly by an inner product B on M by 1 ”
B (v , w) = - C mi(vii wi), 2 1
K(x,v)
=
B(v, v ) .
It is clear that B defines an inner product on M where (vi, w;)means the original inner product on E. The potential energy V is a function on M defined by
We suppose that the gravitational constant is 1 for simplicity. Note that this function is not defined at any “collision” (where x; = zj) . Let Aij be the subspace of collisions of the ith and j t h bodies so that A;j
=
{ x E M I xi
= xj}, i
Thus Aij is a linear subspace of the vector space M . Denote the space of all collisions by A C M so that A = U A + Then properly speaking, the domain of the potential energy is M - A: V : M - A-R.
Q1.
289
THE 11-BODY PROBLEM
We deal then with the space of noncollision states which is ( M - A) X M . Newton’s equations are second order equations on M - A which may be written mifi
=
-gradi V(Z)
for i
=
1,.
. . , 72.
Here the partial derivative D;V of V with respect to z; is a map from M - A to L ( E , R ) ; then the inner product on E convertsD;V(x) to a vector which we call grad; V ( z ) .T he process is similar to the definition of gradient in Chapter 9. Thus the equations make sense as written. One may rewrite Newton’s equations in such a way that they become a first order system on the space of states (A4 - A ) X M :
x; =
vi,
mivj = -grad,V(z),
for i
= , . .
. , n.
The flow obtained from this differential equation then determines how a state moves in time, or the life history of the n bodies once their positions and velocities are given. Although there is a vast literature of several centuries on these equations, no clear picture has emerged. In fact it is still not even clear what the basic questions are for this “problem.” Some of the questions that have been studied include: Is it true that almost all states do not lead to collisions? To what extent are periodic solut,ions stable? How to show the existence of periodic solutions? How to relate the theory of the n-body problem to the orbits in the solar system? Our present goal is simply to put Newton’s equations into the framework of this book and to see how they fit into the more abstract framework of Hamiltonian mechanics. We put the n-body problem into a little more general setting. The key ingredients are : (1)
Configuration space Q, an open set in a vector space E (in the above case
Q = M - AandE = M ) . (2) A C2 function K : Q X E + R, kinetic energy, such that K ( s , v ) has the form K ( z , v ) = K z ( v ,v ) , where K , is an inner product on E (in the above case K , was independent of z, but in problems with constraints, K , de-
pends on 2). (3) A C2 function V : Q + R,potential energy.
The triple,(&, K , V ) is called a simple mechanical system, and Q x I3 the slate apace of the system. Given a simple mechanical system ( Q , K , V ) the energy or total energy is the function e: Q X E + R defined by e(z, v ) = K ( z , v ) V(z). For a simple mechanical system, one can canonically define a vector field on Q x E which gives the equations of motion for the states (points of Q X E). We will see how this can be done in the next section.
+
14.
290
CLASSICAL
MECHANICS
Examples of simple mechanical systems beside the n-body problem include a particle moving in a conservative central force field, a harmonic oscillator, and a frictionless pendulum. If one extends the definition of simple mechanical systems to permit Q to be a manifold, then a large part of classical mechanics may be analyzed in this framework.
$2.
Hamiltonian Mechanics
We shall introduce Hamiltonian mechanics from a rather abstract point of view, and then relate it to the Newtonian point of view. This abstract development proceeds quite analogously to the modern treatment of gradients using inner products; now however the inner product is replaced by a “symplectic form.” So we begin our discussion by defining this kind of form. If F is a vector space, a symplectic form on F is a real-valued bilinear form that is antisymmetric and nondegenerate. Thus 8:FXF+R
is a bilinear map that is antisymmetric: Q(u, v ) = -fl(u, which means that the map +Pa
is an isomorphism. Here
= CP:
u), and nondegenerate,
F + F*
is the linear map from F to F* defined by + ( u ) ( v )= n(u, o ) ,
U, u
E F.
It turns out that the existence of a symplectic form on F implies that the dimension of F is even (see the Problems). We give an example of such a form noon every even dimensional vector space. If F is an even dimensional vector space, we may write F in the form F = E X E*, the Cartesian product of a vector space E and its dual E*. Then an element f of F is of the form (u, 20) where u, ware vectors of E , E*, respectively. Now iff = (u, w), f” = (tP, Up) are two vectors of F, we define flo(f,f”)
=
w“(v>
- w(fl>.
Then it is easy to check that 0, is a symplectic form on F. The nondegeneracy is obtained by showing that if a # 0, then one may find j3 such that %(a, 8) jc 0. Note that (Ilo does not depend on a choice of coordinate structure on E , so that it is natural on E X E l . If one chooses a basis for El and uses the induced basis on E*, nois expressed in coordinates by Q,( (u, w ) , (rp,
Up))
=
c wioui - c
wiuio.
92.
291
HAMILTONIAN MECHANICS
It can be shown that every symplectic form is of this type for some representation of F as E X E*. Now let U be an open subset of a vector space F provided with a symplectic form n. There is a prescription for assigning to any c1 function H : U 4 R,a C1 vector field X H on U called the Hamiltonian vector field of H . In this context H ia called a Humiltonian or a Hamiltonian function. To obtain X H let D H : U 4 F* be the derivative of H and simply write (1)
X,(S)
= W’DH(z),
2
E U,
where 0-’is the inverse of the isomorphism 0: F + F* defined by n above. (1) is equivalent to saying ~ ( X(2) H , y ) = DH (2)( y ) ,all y E F. Thus XH:U + F is a C’ vector field on U ; the differential equations defined by this vector field are called Hamilton’s equations. By using coordinates we can compare these with what are called Hamilton’s equations in physics books. Let no be the symplectic form on F = E X E* defined above and let 2 = (xl, . . . , xn) represent points of E and y = ( y l , . . . , y n ) points of E* for the dual coordinate structures on E and E*. Let 0 0 : F. + F* be the associated isomorphism. For any c1 function H : U + R,
Frorn this, one has that @(;‘DH(x,y) is the vector with components
This is seen as follows. Observe that (suppressing (2,y ) ) or
@ o ( X H ) = DH
no(&,
W)
for all
= DH(w)
w
E F.
By letting w range over the standard basis elements of R*”,one confirms the expreasion for X H . The differential equation defined by the vector field XH is then: 2: =
aH -
ayi ’
i
= 1,
. . . , 72,
These are the usual expressions for Hamilton’s equations. Continuing on the abstract level we obtain the “conservation of energy” theorem. The reason for calling it by this name is that in the mechanical models described
292
14.
CLASSICAL MECHANICS
in this setting, H plays the role of energy, and the solution curves represent the motions of states of the system. Theorem (Conservation of Energy) Let U be an open set of a vector space F , H : U + R any c2 junction and 52 a symplectic form on F . Then H i s constant on the solution curves defined by the vector $eld X H . Proof. If + t ( z ) is a solution curve of the vector field X H , then it has to be shown that
d - H ( t $ , ( z ) ) = 0, dt
all z, t .
This expression by the chain rule is
(a
DH(t$f(z))
+f(z,) = DH(XH).
But DH(XH) is simply, by the definition of XH, 52(XH, XH)which is 0 since 52 is antisymmetric. This ends the proof.
It is instructive to compare this development with that of a gradient dynamical system. These are the same except for the character of the basic bilinear form involved; for one system it is an inner product and for the other it is a symplectic form. The defining function is constant on solution curves for the Hamiltonian case, but except a t equilibria, i t is increasing for the gradient case. From the point of view of mechanics, the Hamiltonian formulation has the advantage that the equations of motion are expressed simply and without need of coordinates, starting just from the energy H . Furthermore, conservation laws follow easily and naturally, the one we proved being the simplest example. Rather than pursue this direction however, we turn to the question of relating abstract Hamiltonian mechanics to the more classical approach to mechanics. We shall see how the energy of a simple mechanical system can be viewed as a Hamiltonian H ; the differential equations of motion of the system are then given by the vector field XH. Thus to a given simple mechanical system ( Q , K , V ), we will associate a Hamiltonian system H : U + R, U C F , Q a symplectic form on F in a natural way. Recall that configuration space Q is an open set in a vector space E and that the state space of the simple mechanical system is Q X E . The space of generalized momenta or phase space of the system is Q X E*, where E* is the dual vector space of Q . The relation between the state space and the phase space of the system is given by the Legendre transformution A : Q X E + Q X E*. To define A, first define a linear isomorphism A,: E + E*, for each q E Q, by A,(u)w
=
2K,(v, w ) ;
u E El
20
E E.
82.
293
HAMILTONIAN MECHANICS
Then set
v)
= (q,
MV)).
Consider the example of a simple mechanical system of a particle with mass m moving in Euclidean three space E under a conservative force field given by potential energy V . In this case state space is E X E and K : E X E + R is given by K ( q , u ) = 3m I u 12. Then A : E X E + E X E* is given by h,(v) = p E E*, where p ( w ) = 2K,(v, t o ) ; or P ( W > = m(v, 2 0 ) and ( , ) is the inner product on E. I n a Cartesian coordinate system on E , p = mu, so that the image p of v under h is indeed the classical momentum, “conjugate” to 2). Returning to our simple mechanical system in general, note that the Legendre transformation has an inverse, so that h is a diffeomorphism from the state space to the phase space. This permits one to transfer the energy function e on state space to a function H on phase space called the Hamiltonian of a simple mechanical system. Thus we have
H
R =
eoX-1
The final step in converting a simple mechanical system to a Hamiltonian system is to put a symplectic form on F = E X E* 3 Q X E* = U . But we have already constructed such a form noin the early part of this section. Using ( q , p ) for variables on Q X E*, then Hamilton’s equations take the form in coordinates
p!
=
dH
-- , aqc
i=l,
. . . , n.
Since for a given mechanical system N (interpreted as total energy) is a known function of p , , q,, these are ordinary diflerential equations. The basic assertion of Hamiltonian mechanics is that they describe the motion of the system. The justification for this assertion is twofold. On one hand, there are many cases where Hamilton’s equations are equivalent to Newton’s; we discuss one below. On the other hand, there are common physical systems to which Newton’s laws do not directly apply (such as a spinning top), but which fit into the framework of “simple mechanical systems,” especially if the configuration space is allowed to be a surface or higher dimensional manifold. For many such systems, Hamilton’s eauations have been verified experimentally.
294
14. CLABBICAL
MECHANIC8
It is meaningless, however, to try to deduce Hamilton’s equations from Newton’s on the abstract level of simple mechanical system (Q,K, V). For there is no identification of the elements of the “configuration space” Q with any particular physical or geometrical parameters. Consider as an example the special case above where K(q, v) = 1 C m,vt in Cartesian coordinates. Then wv,= pr and H(jo, q ) = C h 2 / 2 m r )+ V ( q ); Hamilton’s equations become
Differentiating the first and combining these equations yield
These are the familii Newton’s equations, again. Conversely, Newton’s equations imply Hamilton’s in this case.
PROBLEMS 1. Show that if the vector @ace F
has a symplectic form il on it, then F has even dimension. Hint:Give F an inner product ( , ) and let A : F +F be the operator defined by ( A z , y ) = Q(z, y). Consider the eigenvectors of A.
2. (Lagrange) Let (Q,K, V) be a simple mechnnical system and XH the associated Hamiltonian vector field on phase space. Show that (qJ 0) is an equilibrium for XH if and only if DV(q) = 0 ; and (qJ0 ) is a stable equilibrium if q is an isolated minimum of V. (Hint: Use conservation of energy.) 3. Consider the second order differential equation in one variable
3 +f(z) = 0’
where f: R --* R is ?t and if f(z) = 0, then f’( z) # 0. Describe the orbit structure of the associated system in the plane J=v
8 = -f(z)
when f(z) = z - 9. Discuss this phase-portrait in general. (Hid: Consider
H(zJv,
=
3@
+f 0
f ( t ) dt
295
NOTES
and show that H is constant on orbits. The critical points of H are at v = 0, f(z) = 0; use H,, = f’(z), H , , = 1.) 4.
Consider the equation
I
+ g ( z ) k + f(z)
=
0,
where g(z) > 0, and f is as in Problem 3. Describe the phase portrait (the function y may be interpreted as coming from friction in a mechanical problem).
Notes
One modern approach to mechanics is Abraham’s book, Foundations of Mechanics [l]. Wintner’s Analytical Foundations of Celestial Mechanics C25) has a very extensive treatment of the n-body problem.
14 Chssical Mechanics
The goal of this very short chapter is to do two things: (1) to give a statement of the famous n-body problem of celestial mechanics and (2) to give a brief introduction to Hamiltonian mechanics. We give a more abstract treatment of Hamiltonian theory than is given in physics texts; but our method exhibits invariant notions more clearly and has the virtue of passing easily to the case where the configuration space is a manifold. $1. The n-Body Problem
We give a description of the n-body “problem” of celestial mechanics; this extends the Kepler problem of Chapter 2. The basic example of this mechanical system is the solar system with the sun and planets representing the n bodies. Another example is the system consisting of the earth, moon, and sun. We are concerned here with Newtonian gravitational forces; on the other hand, the Newtonian n-body problem is the prototype of other n-body problems, with forces other than gravitational. The data, or parameters of this system, are n positive numbers representing the masses of the n bodies. We denote these numbers by ml, . . . , m.. The first goal in understanding a mechanical system is to define the configuration space, or space of generalized positions. In this case a configuration will consist precisely of the positions of each of the n bodies. We will write xi for the position of the ith body so that z i is a point in Euclidean three space (the space in which we live) denoted by E. Now E is isomorphic to Cartesian space R*but by no natural isomorphism. However E does have a natural notion of inner product and associated norm; the notions of length and perpendicular make sense in the space in which we live, while any system of coordinate axe8 is arbitrary.
288
14.
CLASSICAL MECHANICS
Thus Euclidean three space, the configuration space of one body, is a threedimensional vector space together with an inner product. The configuration space M for the n-body problem is the Cartesian product of E with itself n times; thus M = ( E ) ”and x = ( 5 1 , . . . , xn), where xi E E is the position of the ith body. Note that x, denotes a point in El not a number. One may deduce the space of states from the configuration space as the space TM of all tangent vectors to all possible curves in M . One may think of T M as the product M X M and represent a state as ( x , v ) E M X M , where x is a configuration as before and v = (vl, . . , vn), v i E E being the velocity of the ith body. A state of the system gives complete information about the system at a given moment and (at least in classical mechanics) determines the complete life history of the state. The determination of this life history goes via the ordinary differential equations of motion, Newton’s equations in this instance. Good insights into these equations can be obtained by introducing kinetic and potential energy. The kinetic energy is a function K : M X M + R on the space of states which is given by
.
~ ( 2 V ,)
=
’c 2
mi
i-1
I v i 12.
Here the norm of v ; is the Euclidean norm on E. One may also consider K to be given directly by an inner product B on M by 1 ”
B (v , w) = - C mi(vii wi), 2 1
K(x,v)
=
B(v, v ) .
It is clear that B defines an inner product on M where (vi, w;)means the original inner product on E. The potential energy V is a function on M defined by
We suppose that the gravitational constant is 1 for simplicity. Note that this function is not defined at any “collision” (where x; = zj) . Let Aij be the subspace of collisions of the ith and j t h bodies so that A;j
=
{ x E M I xi
= xj}, i
Thus Aij is a linear subspace of the vector space M . Denote the space of all collisions by A C M so that A = U A + Then properly speaking, the domain of the potential energy is M - A: V : M - A-R.
Q1.
289
THE 11-BODY PROBLEM
We deal then with the space of noncollision states which is ( M - A) X M . Newton’s equations are second order equations on M - A which may be written mifi
=
-gradi V(Z)
for i
=
1,.
. . , 72.
Here the partial derivative D;V of V with respect to z; is a map from M - A to L ( E , R ) ; then the inner product on E convertsD;V(x) to a vector which we call grad; V ( z ) .T he process is similar to the definition of gradient in Chapter 9. Thus the equations make sense as written. One may rewrite Newton’s equations in such a way that they become a first order system on the space of states (A4 - A ) X M :
x; =
vi,
mivj = -grad,V(z),
for i
= , . .
. , n.
The flow obtained from this differential equation then determines how a state moves in time, or the life history of the n bodies once their positions and velocities are given. Although there is a vast literature of several centuries on these equations, no clear picture has emerged. In fact it is still not even clear what the basic questions are for this “problem.” Some of the questions that have been studied include: Is it true that almost all states do not lead to collisions? To what extent are periodic solut,ions stable? How to show the existence of periodic solutions? How to relate the theory of the n-body problem to the orbits in the solar system? Our present goal is simply to put Newton’s equations into the framework of this book and to see how they fit into the more abstract framework of Hamiltonian mechanics. We put the n-body problem into a little more general setting. The key ingredients are : (1)
Configuration space Q, an open set in a vector space E (in the above case
Q = M - AandE = M ) . (2) A C2 function K : Q X E + R, kinetic energy, such that K ( s , v ) has the form K ( z , v ) = K z ( v ,v ) , where K , is an inner product on E (in the above case K , was independent of z, but in problems with constraints, K , de-
pends on 2). (3) A C2 function V : Q + R,potential energy.
The triple,(&, K , V ) is called a simple mechanical system, and Q x I3 the slate apace of the system. Given a simple mechanical system ( Q , K , V ) the energy or total energy is the function e: Q X E + R defined by e(z, v ) = K ( z , v ) V(z). For a simple mechanical system, one can canonically define a vector field on Q x E which gives the equations of motion for the states (points of Q X E). We will see how this can be done in the next section.
+
14.
290
CLASSICAL
MECHANICS
Examples of simple mechanical systems beside the n-body problem include a particle moving in a conservative central force field, a harmonic oscillator, and a frictionless pendulum. If one extends the definition of simple mechanical systems to permit Q to be a manifold, then a large part of classical mechanics may be analyzed in this framework.
$2.
Hamiltonian Mechanics
We shall introduce Hamiltonian mechanics from a rather abstract point of view, and then relate it to the Newtonian point of view. This abstract development proceeds quite analogously to the modern treatment of gradients using inner products; now however the inner product is replaced by a “symplectic form.” So we begin our discussion by defining this kind of form. If F is a vector space, a symplectic form on F is a real-valued bilinear form that is antisymmetric and nondegenerate. Thus 8:FXF+R
is a bilinear map that is antisymmetric: Q(u, v ) = -fl(u, which means that the map +Pa
is an isomorphism. Here
= CP:
u), and nondegenerate,
F + F*
is the linear map from F to F* defined by + ( u ) ( v )= n(u, o ) ,
U, u
E F.
It turns out that the existence of a symplectic form on F implies that the dimension of F is even (see the Problems). We give an example of such a form noon every even dimensional vector space. If F is an even dimensional vector space, we may write F in the form F = E X E*, the Cartesian product of a vector space E and its dual E*. Then an element f of F is of the form (u, 20) where u, ware vectors of E , E*, respectively. Now iff = (u, w), f” = (tP, Up) are two vectors of F, we define flo(f,f”)
=
w“(v>
- w(fl>.
Then it is easy to check that 0, is a symplectic form on F. The nondegeneracy is obtained by showing that if a # 0, then one may find j3 such that %(a, 8) jc 0. Note that (Ilo does not depend on a choice of coordinate structure on E , so that it is natural on E X E l . If one chooses a basis for El and uses the induced basis on E*, nois expressed in coordinates by Q,( (u, w ) , (rp,
Up))
=
c wioui - c
wiuio.
92.
291
HAMILTONIAN MECHANICS
It can be shown that every symplectic form is of this type for some representation of F as E X E*. Now let U be an open subset of a vector space F provided with a symplectic form n. There is a prescription for assigning to any c1 function H : U 4 R,a C1 vector field X H on U called the Hamiltonian vector field of H . In this context H ia called a Humiltonian or a Hamiltonian function. To obtain X H let D H : U 4 F* be the derivative of H and simply write (1)
X,(S)
= W’DH(z),
2
E U,
where 0-’is the inverse of the isomorphism 0: F + F* defined by n above. (1) is equivalent to saying ~ ( X(2) H , y ) = DH (2)( y ) ,all y E F. Thus XH:U + F is a C’ vector field on U ; the differential equations defined by this vector field are called Hamilton’s equations. By using coordinates we can compare these with what are called Hamilton’s equations in physics books. Let no be the symplectic form on F = E X E* defined above and let 2 = (xl, . . . , xn) represent points of E and y = ( y l , . . . , y n ) points of E* for the dual coordinate structures on E and E*. Let 0 0 : F. + F* be the associated isomorphism. For any c1 function H : U + R,
Frorn this, one has that @(;‘DH(x,y) is the vector with components
This is seen as follows. Observe that (suppressing (2,y ) ) or
@ o ( X H ) = DH
no(&,
W)
for all
= DH(w)
w
E F.
By letting w range over the standard basis elements of R*”,one confirms the expreasion for X H . The differential equation defined by the vector field XH is then: 2: =
aH -
ayi ’
i
= 1,
. . . , 72,
These are the usual expressions for Hamilton’s equations. Continuing on the abstract level we obtain the “conservation of energy” theorem. The reason for calling it by this name is that in the mechanical models described
292
14.
CLASSICAL MECHANICS
in this setting, H plays the role of energy, and the solution curves represent the motions of states of the system. Theorem (Conservation of Energy) Let U be an open set of a vector space F , H : U + R any c2 junction and 52 a symplectic form on F . Then H i s constant on the solution curves defined by the vector $eld X H . Proof. If + t ( z ) is a solution curve of the vector field X H , then it has to be shown that
d - H ( t $ , ( z ) ) = 0, dt
all z, t .
This expression by the chain rule is
(a
DH(t$f(z))
+f(z,) = DH(XH).
But DH(XH) is simply, by the definition of XH, 52(XH, XH)which is 0 since 52 is antisymmetric. This ends the proof.
It is instructive to compare this development with that of a gradient dynamical system. These are the same except for the character of the basic bilinear form involved; for one system it is an inner product and for the other it is a symplectic form. The defining function is constant on solution curves for the Hamiltonian case, but except a t equilibria, i t is increasing for the gradient case. From the point of view of mechanics, the Hamiltonian formulation has the advantage that the equations of motion are expressed simply and without need of coordinates, starting just from the energy H . Furthermore, conservation laws follow easily and naturally, the one we proved being the simplest example. Rather than pursue this direction however, we turn to the question of relating abstract Hamiltonian mechanics to the more classical approach to mechanics. We shall see how the energy of a simple mechanical system can be viewed as a Hamiltonian H ; the differential equations of motion of the system are then given by the vector field XH. Thus to a given simple mechanical system ( Q , K , V ), we will associate a Hamiltonian system H : U + R, U C F , Q a symplectic form on F in a natural way. Recall that configuration space Q is an open set in a vector space E and that the state space of the simple mechanical system is Q X E . The space of generalized momenta or phase space of the system is Q X E*, where E* is the dual vector space of Q . The relation between the state space and the phase space of the system is given by the Legendre transformution A : Q X E + Q X E*. To define A, first define a linear isomorphism A,: E + E*, for each q E Q, by A,(u)w
=
2K,(v, w ) ;
u E El
20
E E.
82.
293
HAMILTONIAN MECHANICS
Then set
v)
= (q,
MV)).
Consider the example of a simple mechanical system of a particle with mass m moving in Euclidean three space E under a conservative force field given by potential energy V . In this case state space is E X E and K : E X E + R is given by K ( q , u ) = 3m I u 12. Then A : E X E + E X E* is given by h,(v) = p E E*, where p ( w ) = 2K,(v, t o ) ; or P ( W > = m(v, 2 0 ) and ( , ) is the inner product on E. I n a Cartesian coordinate system on E , p = mu, so that the image p of v under h is indeed the classical momentum, “conjugate” to 2). Returning to our simple mechanical system in general, note that the Legendre transformation has an inverse, so that h is a diffeomorphism from the state space to the phase space. This permits one to transfer the energy function e on state space to a function H on phase space called the Hamiltonian of a simple mechanical system. Thus we have
H
R =
eoX-1
The final step in converting a simple mechanical system to a Hamiltonian system is to put a symplectic form on F = E X E* 3 Q X E* = U . But we have already constructed such a form noin the early part of this section. Using ( q , p ) for variables on Q X E*, then Hamilton’s equations take the form in coordinates
p!
=
dH
-- , aqc
i=l,
. . . , n.
Since for a given mechanical system N (interpreted as total energy) is a known function of p , , q,, these are ordinary diflerential equations. The basic assertion of Hamiltonian mechanics is that they describe the motion of the system. The justification for this assertion is twofold. On one hand, there are many cases where Hamilton’s equations are equivalent to Newton’s; we discuss one below. On the other hand, there are common physical systems to which Newton’s laws do not directly apply (such as a spinning top), but which fit into the framework of “simple mechanical systems,” especially if the configuration space is allowed to be a surface or higher dimensional manifold. For many such systems, Hamilton’s eauations have been verified experimentally.
294
14. CLABBICAL
MECHANIC8
It is meaningless, however, to try to deduce Hamilton’s equations from Newton’s on the abstract level of simple mechanical system (Q,K, V). For there is no identification of the elements of the “configuration space” Q with any particular physical or geometrical parameters. Consider as an example the special case above where K(q, v) = 1 C m,vt in Cartesian coordinates. Then wv,= pr and H(jo, q ) = C h 2 / 2 m r )+ V ( q ); Hamilton’s equations become
Differentiating the first and combining these equations yield
These are the familii Newton’s equations, again. Conversely, Newton’s equations imply Hamilton’s in this case.
PROBLEMS 1. Show that if the vector @ace F
has a symplectic form il on it, then F has even dimension. Hint:Give F an inner product ( , ) and let A : F +F be the operator defined by ( A z , y ) = Q(z, y). Consider the eigenvectors of A.
2. (Lagrange) Let (Q,K, V) be a simple mechnnical system and XH the associated Hamiltonian vector field on phase space. Show that (qJ 0) is an equilibrium for XH if and only if DV(q) = 0 ; and (qJ0 ) is a stable equilibrium if q is an isolated minimum of V. (Hint: Use conservation of energy.) 3. Consider the second order differential equation in one variable
3 +f(z) = 0’
where f: R --* R is ?t and if f(z) = 0, then f’( z) # 0. Describe the orbit structure of the associated system in the plane J=v
8 = -f(z)
when f(z) = z - 9. Discuss this phase-portrait in general. (Hid: Consider
H(zJv,
=
3@
+f 0
f ( t ) dt
295
NOTES
and show that H is constant on orbits. The critical points of H are at v = 0, f(z) = 0; use H,, = f’(z), H , , = 1.) 4.
Consider the equation
I
+ g ( z ) k + f(z)
=
0,
where g(z) > 0, and f is as in Problem 3. Describe the phase portrait (the function y may be interpreted as coming from friction in a mechanical problem).
Notes
One modern approach to mechanics is Abraham’s book, Foundations of Mechanics [l]. Wintner’s Analytical Foundations of Celestial Mechanics C25) has a very extensive treatment of the n-body problem.