- Email: [email protected]
A direct construction of first integrals for certain non-linear dynamical systems
A DIRECT CONSTRUCTION FOR CERTAIN NON-LINEAR
OF FIRST INTEGRALS DYNAMICAL SYSTEMS
w. SARLET Instituut voor Theoretische Mechanica Rijksuniversiteit Gent, Krijgshtan 27169, B-9000 Gent, Belgium and L. Y. BAHARt Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, U.S.A. (Received S October 1979) Abstract-A direct, constructive approach to the problem of finding first integrals of certain nonlinear, second order ordinary differential equations is presented. The idea is motivated by the construction of the energy integral for the equations of motion of the corresponding conservative systems. Although the method developed for the class of equations studied herein is elementary, it yields the same results as the more advanced group-theoretical methods, such as the use of symmetries in the context of Noether’s theorem. The approach reveals some interesting features when it is specialized to the case of linear equations. Finally, a tw~dimensional example is considered by extending the methodology developed for scalar equations to their vector counterparts. It is shown that, as a consequence. a first integral which is independent of the energy integral exists for a particular HamiItonian of the Contopoulos type.
1. INTRGDUCT~ON
The importance of finding first integrals for ordinary differential equations has been recap nized since the earliest developments in this subject. A complete set of first integrals determines the solution of a system of differential equations. It is even more important to find a first integral in cases where it is difficult, if not impossible, to find explicit solutions, since the determination of an integral reduces the order of the differential equation, and often gives qualitative information about the behavior of the underlying dynamical system. Despite long-standing interest and research, there stil1 appear many papers emphasi~ng various approaches to derive first integrals for those cases that are not amenable to a straightforward treatment. These cases consist for example of dynamical systems governed by nonconservative forces, that give rise for the most part, to non-linear differential equations with time-dependent coefficients. The most powerful methodology by which this problem can be approached is based on group-theoretica considerations, which relate first integrals to the symmetries of the system. Moreover, a knowledge of the symmetry is often as valuable as knowing the resulting invariant. There are two closely related approaches in these methods. The first is based on the theory of the symmetries of Hamiltonian systems, generated by one-parameter families of infinitesimal canonical transformations, a recent treatment of which can be found in fl]. The other is the Noether theorem and its various generalizations. Among the many papers that have been published on this subject, we cite a few of the more recent ones [2-53, as well as the text by Logan [6]. These ideas have found their highest degree of generalization with the introduction of the concept of momentum mapping in sympiectic geometry which is described in detail in [7]. The application of Noether-type methods to practical problems require the solution of a system of partial differential equations, known as the Killing equations, or some general*Research supported by a Nato grant. tResearch supported by the U.S. Department of Energy under contracts ET-78-C-01-2092, 3088. 133
and ET-78-S-01-
134
W. S.wmand
L. Y. BAHAR
ization of those equations. Applications to various examples of the Killing equations are to be found in [2, S-11]_ In order to apply the above methods one has to know a priori a Lagrangian or ~amiltonian describing the system, or alternatively a more general variational principle such as the one described in [9]. The general theory of Lie symmetry groups, however, can be applied directly to any differential equation as indicated in [12-141. As is well known, this procedure enables one to determine the symmetries of the equation, which are not always necessarily related to first integrals [lo]. Some authors have tried to avoid the sophistication of group-theoretical methods, and have developed various ad hoc approaches for the ~nst~ction of first integrals. Al~ough such approaches are less general than group-theoretical ones, they can lead to the desired result more directly for specific problems, thus bypassing the necessity for solving simultaneous partial differential equations altogether. An example of such a method is the transformation approach proposed by Crespo da Silva [15-J. The idea behind this method is to search for a canonical transformation which reduces a non-conservative Hamiltonian to the product of a function of time, and a function of phase-space variables only, This latter function produces a constant of the motion. The idea developed in [15] is, up to this point, quite general. However, it is readily seen that it is an ad hoc approach, because only linear canonical transformations are considered, and that no information is available about the type of Hamiltonians which can be cast into the desired form by such transformations. Another area in which the generation of first integrals becomes necessary is the stability analysis of dynamical systems through the use of the second method of Lyapunov. As representative contri&tions to this field one may cite the recent work due to Djukic [16, 173, and Risito [lg]. It is, in part. to generalize existing studies to encompass the stability of linear non-conservative dynamical systems that the present work was undertaken. The purpose of this paper is to present a new direct approach to the instruction of a first integral for second order Newtonian systems. The problem is approached directly at the level of the equation of motion as given, without the need of going through a Lagrangian or Hamiltonian reformulation. Roughly speaking, the method consists of trying to construct a first integral in a manner similar to the one used in obtaining the energy integral for conservative systems, namely that of multiplying the equation of motion by an appropriate integrating factor. The present study covers all the results obtained in [15], and in addition, the constructive procedure permits the identification of the class of equations that can be treated by the method developed here. It is shown that for this class of equations the results are in agreement with those obtained from a direct application of Noether’s theorem. Furthermore, one gains new insight about the peculiarities of the linear case, which explains why the invariant of the time-dependent harmonic oscillator, found by Lewis [19] cannot be expected to be generalized to many non-linear cases. Finally, the ideas behind the present method are extended to systems with multiple degrees of freedom, in which an example of Hamiltonians studied by Contopoulos [20] is treated. The results reveal that, for a special case, a constant of the motion, which differs from the Hamiltonian, can be exhibited in closed-form. Finally, possible extensions, which are currently under study, are indicated.
2. INVARIANT
FOR
THE
EMDEN
EQUATION
Consider the equation
(1) which is known in the literature as the Emden equation [6, pp. 52-541. Following Jones and Ames [21], we eliminate the middle term of equation(l), by introducing the transformation 4 = U(r)&
(2)
A direct construction
of first integrals for certain non-linear
dynamical systems
135
which gives rise to the equation Ui+2(ti+q)i+(ti+;o)*+“Jx~=0.
(3)
Clearly, the choice u= l/t eliminates the middle term of equation (3) and simplifies it into X5 it+-=o, t4
(4)
which is the reduced form of the Emden equation. We now attempt to integrate equation (4) by multiplying it with h(t)& following the usual method employed for conservative systems, as a non-conservative term has already been eliminated, but allowing in addition the possibility of a time-dependent integrating factor. In the resulting equation, &)x2+?
x5x=0,
both terms can be written as a total time-derivative the time-dependent coefficient. We get, ;
[+h(t){i2++t-4X6}]
-$h(t)k2 -
(5)
with a correction for the derivative of
$ [h(t)~-~] f
=o.
The last two terms cannot be combined to a total time-derivative as they stand. We therefore appeal to the differential equation for replacing one of these terms. Multiplying equation (4) by x, we have x6 = _ tbxji,
(7)
which can be used in the last term of equation (6) to yield,
It becomes clear that we now have the possibility of writing the last three terms in (8) as the time-derivative of a function likef(t)xx, namely as g
[f(t)xi]
=j(t)xk +f(t)[i2 + xi].
Identification of the right hand side of equation (9) with the corresponding terms in equation (8) leads to the relations, f(c)= -$(t), (lOa) j(t)=O,
(lob)
f(t)=~[ri(t)-4t-‘h(t)],
(1Oc)
which clearly admit the solution h(t) = t,
f(t)= -4.
The above choice enables us to write the first integral of equation (4) in the form X6 tX2 + 3t3
-xi
=
constant,
(11)
which is the conservation relation derived by Jones and Ames [21] using similarity variables. It is seen that the ad hoc nature of the method resides in the fact that one must find two functions satisfying three relations, equation (lo), which is not always possible. Looking carefully at the structure of the terms produced in the steps leading to equations (6) and (9) it is also seen that the method can be extended to more general cases. As a matter of fact, it can be applied directly to the Emden equation in its original form (1). Proceeding in the same manner we obtain (12)
136
W. SARLETand L. Y. BAHAR
which we rewrite first in the form, $ [~h(c){~~+~~}]+[2t-~h(t)-~~(t)]~~-~~(t)q~=o,
(13)
and subsequently, using the differential equation to replace the last term, as $ [-/z(t){Cj*+~q6}]+ [2t-‘h(t)-~~(c)]q* Again, we can attempt to find a functionf(t) become equal to $
-+&t)[q~+2PqCj]=O.
(14)
such that the last three terms in equation (14)
CCrhmk4l =md +_m[4*+ Ml.
(15)
This gives rise to three equations for the unknown functions h(t),f(t), having as a solution h(t) = t3,
f(t)=fP.
So we obtain the first integral
t3(Q2+$I’) + t*qcj = constant,
(16)
which of course also follows from (11) and (2). We have treated this example in great detail, in order to recognize the essential features which make the approach succeed. Important is the linearity of the equation in the derivatives, while the non-linearity in q must be of polynomial character, and remain so after integration. The coefficients can be general functions of c, but they will have to satisfy a certain compatibility relation. This is clear from the fact that at the final stage, we obtained a system of equations like (lo), which in general will be overdetermined.
3. FIRST
INTEGRAL
FOR
A CLASS
OF
NON-LINEAR
EQUATIONS
From the considerations of Section 2, we conclude that a first integral will exist for the following class of equations, fj + /I(t)4 + a(t)q” = 0,
(17)
m#-1,
under certain restrictions on u and /I. We will derive these restrictions by the direct approach to the construction integral. Multiplying (17) by 2Y(t)& where y is yet undetermined, we get
of a first
2yqi + 2yb4* + 2yaqmQ = 0, which by manipulating the first and last term can be written as g
(
ul*+&
wf+’
+ 4*(2YB-Y) - --&
>
-$ (ya)q”” =o.
(18)
Making use of the differential equation for the last term, this yields
$
(
r4*+&
wT+’
>
+4’(zrS-t)+--4$ a-l z(yaXqij+Bqq)=O,
(19)
so we will find an invariant, if the remaining terms can be combined into ; (a(t)q~)=a(ri*+qij)+ciqq.
(20)
Identifying the coefficients of q4,4*, and q& we obtain the system of equations a=2y/3-Y, 2
a=--a-l m+l
(21a) -$ya),
(21b)
A direct construction
offirst integrals
for certain non-linear dynamical systems
137
(214
From (21a) and (21b), we get
m+3.
m+l’+m+l
2
y[a-‘dr-(m+
l)B]=O,
which for m # - 3 can be integrated to y=cta -2”mi3)exp(Z~IIB(I’)d~),
(22)
c, being a constant. On the other hand, from (21b) and (21~) we easily obtain a=c2 exp(,’
j?(i.)dt.).
(23)
Finally, using (22) and (23) in (21a), we get a-2/(“+3)(2/?+a-1ci) exp (2 where c3 is a constant, in fact c3 =
I’B(r)df.)=c3.
(m+3)C,
2c,
*
Equation (24) is a sufficient condition to be satisfied by the given coefficients a and fi, in order to yield a first integral. Making the substitution y(t)=a-2/(m+3),
it can be further integrated to a-2jtm+3)exp(
- --$-3 s’
jI(r)dr’)-e,
r exp (-I’
j?(r~)dr~)dr’=c,
(25)
where cq and c are arbitrary constants, where cg= -c&z,. When a and fl satisfy (25), we can compute y and a, and obtain a first integral from (19). We can choose ci = 1, ct = -c,; then the first integral is given by
42+m;l _
fl+l
a-2/(m+3)
exp(2%1)/3(r’)dr’)-c,q~exp(f jI(r’)dt.)=constant,
>
which, making use of (25), can equivalently be written as, (d2+s
q’““)
exp (2
s’fl(t.)dt')[c+c,
r exp (-I” -c&
/?(f.)df.)dt’]
exp (,‘B(r)d~‘)=constant.
(26)
Note that the present straightforward approach has given more general results than the ones obtained in [ 151. Indeed, the results mentioned in [ 153 correspond to the special case c = 0 in equations (25) and (26). Now, one might wonder how our results compare to those obtainable by the Noether approach. Let us first remark, however, that there is an additional degree of freedom in our direct method. Indeed, in the preceding calculations we tried to match the remaining terms of equation (19) by the derivative of a term like a(r)qd, as in equation (20). But, in a more general way, we could have considered a function of the form a(t)qci + b2,
(27)
with k in general constant. In the appendix we show that the results (25,26) are equivalent to those obtained by the gauge-invariant Noether approach, while it is easy to complete
138
W. SARLET and L. Y. BAHAR
the picture by showing that allowing for gauge-variance in the Noether approach, precisely corresponds to the extension (27). We now make the following two observations. Remark 1
For completeness, apart from the a priori excluded case m = - 1, we have to treat the case m= -3 separately. In this instance, equations (21a, b) immediately give rise to a condition relating a and /I, namely ci-2pa = 0,
which integrates into a = cl exp ( 2 s’ B(t’) dr’),
while the solution for a(t) and v(t) is obtained as, a=c2 exp (,’ j?(rf)dr’), y=exp(z
rfl(t’)dtj[c+c,
~exp(_,“jI(tP)dt*)dt’],
and the expression for a first integral then immediately follows from (19) and (20),
1
’ [a(t’)]-1’2 dt’ +c,a”2qo=constant,
(d2
(28)
with c1 = c2c:12. Remark 2
Let us return now to the sufficient condition (25), which ensures the existence of a first integral for equation (17). In the special case where the system is linear, and contains no damping term, i.e. /I=O, m= 1, we obtain the following condition on a(r), a(t)=(c+c,t)-2.
(29)
It is not difficult to find examples of functions a(t) which are not of the form (29), and yet allow the construction of an invariant. One such example is provided by the equation
for which an invariant is given by, as worked out in [14,19-J, t2d2 -2tqq+(tm2+
l)q2=constant.
(31)
This would seem to indicate, that the present ad hoc approach has shortcomings. As we will explain in the next section, however, the linear case, and only the linear case, allows for an interesting additional freedom in the method, which will result in the fact that in principle, an invariant can always be constructed.
4. THE
CASE
OF
A LINEAR
EQUATION
For simplicity, we treat here only the time-dependent by the equation ~+CI?(t)q=O,
harmonic oscillator, governed (32)
but a time-dependent, linear damping term can be included without significant complications. We start our attempt for the direct construction of an invariant in the usual way by multiplying (32) with 2y(t)& 2y4q + 2yw%jq = 0.
(33)
A direct construction
of first integrals for certain non-linear
dynamical systems
139
As before, we cast the first term of (33) into the form
If we follow literally the previous procedure, we would also cast the last term into the form
and eliminate the q2-term by using the differential equation. This usually leaves us with terms in qij, q(i and d2, which generically come from the timederivative of a term in 44. But in this linear case, the last term of (33) already has the structure of one of these remaining terms, which means that we need not absorb it entirely into a total time-derivative as in (35). In turn, this implies that we could have taken any function oft as the coefficient of q2 in producing terms like (35). In other words, we can simply add zero to the equation, by adding and subtracting equivalent terms. Hence we rewrite equation (33) in the form,
and make use of the differential equation to eliminate the last term, (36) Now again, we try to combine the last three terms of (36) into the form (37) which gives rise to the identifications, 4= -$9
(38a)
+&O-2,
(38b)
I$= 2(yw2 -A).
(38c)
We see that the ‘artifice of adding zero’, which only applies for the linear case, has produced this time a system of three equations with three unknowns (y, J, 4), and hence in principle is always solvable. Note that an alternative way of looking at the peculiarities of the linear case, consists in saying that for m= 1 in equation (17), k can be allowed to depend on t in the extension (27). From (38a, b) we get, 1= -&j,
(39)
while eliminating $J between (38a) and (38~) gives, rz=$+yo? A third-order
(40)
equation in y now follows from elimination of L between (39), and (40), $7 + 205 + 2ywci,= 0.
(41)
This equation coincides with the one obtained by Lutz&y in [ll]. It can be reduced to a second-order equation in the following way. First, we make the substitution Y’P2,
(42)
which allows rewriting equation (41) in the form p $ (li+w2p)+3p(&5+02p)=o.
(43)
Equation (43) has the integrating factor p2, from which it easily follows that p must satisfy
W. SARLETand L. Y. BAHAR
140
the second-order
differential equation cp-3,
b+o2p=
C constant.
(44)
An invariant can now be calculated from (36) and (37). Moreover, the functions y, 1 and 4 can all be expressed in terms of the single function p using (42) and (38). In view of (44), this invariant finally takes the form, Cp-2q2+(pq-~q)2=constant.
(45)
For C= 1, equation (44) and (45) are precisely the results obtained by Lewis [19]. These results have been shown to be important, for instance, for a quantum-mechanical treatment ofthe time-dependent oscillator [22]. They have also been generalized to multi-dimensional but linear systems by Leach [23-25-j, by making use of linear canonical transformations. Other studies aimed at generalizing the Lewis-invariant to non-linear equations have been performed by Symon [26], Sarlet [27], and Leach [28]. For additional aspects, [14] and [29] can also be consulted. Attempts for finding similar results for non-linear equations have not been very successful as can be seen from the complicated expansion procedures in [28]. From the present heuristic approach, we have learned, that the linearity of the equation plays a crucial role for deriving the Lewis-invariant, and that for non-linear problems, the ultimate system of differential equations becomes overdetermined, which means that results will only be found for specific choices of the time-dependent coefficients.
5. A MULTIDIMENSIONAL
NON-LINEAR
EXAMPLE
Consider the dynamical system described by the Hamiltonian (46) b and c being constants. It corresponds to the coupled system of non-linear second-order equations, ci’i+q,+2bq,qz=O, d2fq2+bq:+3cq;=0,
(47)
ij+q+Q(qh=O,
(48)
which we write in matrix form as
where q = col(q,, q2) and (49) Hamiltonians of type (46) have been discussed by Contopoulos [20], and later on by many others, with regard to the existence of a first integral independent from the Hamiltonian. For the special choice b= 1, c= -l/3, (46) gives the Hamiltonian introduced by Htnon and Heiles [30]. In the spirit of the constructive approach we have used to find a first integral for a scalar equation, we start by multiplying the vector-equation (48) on the left by 20%), where V is a possibly time-dependent pose. We get,
v=v’,
2 by 2 matrix, and the superscript T denotes the trans-
24TV4 + 2cfvq + 2ijTw2(q)q = 0, or (50) In this last reformulation,
we have used the insight gained in the previous section, that is,
A direct construction
of first integrals
for certain
non-linear
dynamical
141
systems
we have added zero to the equation through the terms involving W, which is a possibly time-dependent symmetric matrix. The reason for this procedure, is due to the presence of the linear term in q in the given equation (48). There is no loss in generality in assuming W to be symmetric, since qTAq identically vanishes when A is skew-symmetric. We put, V(t) =
40 PO) D(t)- r(t) ’
(
>
(51)
then the last term of equation (50), when computed explicitly gives, 2QTVM)q=4@q,q,9,
+2ybq:q’,+6Bc&,
+4Pq,q,&+2Bbq:&
+%&j,.
(52)
Following the procedure outlined for scalar equations, we normally would combine these terms into the derivative of some polynomial in q with time-dependent coefficients, then make a correction for the derivatives of these coefficients, and finally replace these correction terms by making use of the differential equation. The important point is that these correction terms need to be of polynomial character in q, that is, free of derivatives. This is always automatically the case for a single equation, but it is clear that for multidimensional systems, it can impose additional restrictions, which are essentially due to the coupling of the equations. Roughly speaking, we need to require the terms (52) to have the structure of an exact differential in q, and q2, but not necessarily with the correct coefficients. Inspection of (52) shows that, for the case at hand, we need the requirements, ab= yb,
(53)
/3c=$b.
(54)
Indeed, under these assumptions, we can write 24’VfJ(q)q=$
(2abq:q,+Zflbq$q,
+$3bq:
+2ycq:)-2cibqfq, -2Sbq:q,
-$jbq:
-2vcq;,
(55)
and the equation of motion can be invoked to replace the four correction terms. For b = 0, the equations in q1 and q2 are uncoupled, and one easily finds two constants of the motion, corresponding to the two uncoupled parts of the Hamiltonian. we further need only to investigate the case b#O. Then the correction terms in (55) can be written as -qTr(t)r2(q)q,
with r(t)=
f-2_
ls
and can be replaced from the equation of motion (48) by, - qTrR(q)q = $I-4 + qTrq. So, going back to equation (50), and taking into account the manipulations on the last term, we get
$ (tjTVtj + qTWq + 2abqfq, + Vbqiq,
+ $bq:
we performed
+ 2vcqz)
-qTVq+2tjT(v-w)q-qT(vL-r)q+q’rij=o.
(57)
In this manner we reach the final step in the approach, which is that in order to find a first integral the remaining non-derivative terms in (57) should come from, $ 6jTD0)B)= cjTD(t)Q+ qTD(t)ii + qTD(t)Q, D being an arbitrary time-dependent equations (57) and (58) we get,
matrix. Identifying
the corresponding
(58)
terms in
D(t) = r(t),
(59a)
D(t)= -%‘(t)+A,(t),
(59b)
142
W. SARLETand L. Y. BAHAR
w =Wt)
(59d
- WO),
I-(t)= Ii’(t) + A*(t),
(59d)
where A,(t), and A&) are arbitrary skew-symmetric matrices. From (59a, b, d) we get Ki’+A,= -%‘+A,, and, recalling that V and W are symmetric, we conclude that W(r)= -V(r),
60)
A2(r)=Al(r), which we will denote by A(r).
(61)
Furthermore, equation (59~) shows that fi must be symmetric, which in view of (59b) implies that A should be constant. Now, from (53) and (54), and in view of the fact that b #O, we get a = y, and either /?=O or c= b/3. Hence, two cases have to be considered separately. Case 1 a=y,
p=o.
(62)
We then have
“=(1;I), r$
;).
(63)
Since from (59a, b) the symmetric part of r should equal -V, we conclude that a must be constant, and therefore we have, &D=A,=A,=V=W=(),
(64)
while from (59c), we get v=w.
(65)
From (57) and (58), we conclude that a first integral is given by qTVq + qTVq + 2a(bqtq, + cqz)= constant,
(66)
which in view of the nature of V is nothing else but a multiple of the energy integral, and therefore does not produce a new result. Case 2 a=y,
c = b/3.
(67)
We now have
v=
(a p) B a’
1-+.
For the same reason as before, we conclude that r must be zero, which implies (64) and (65), and of course that a and p are constant, but arbitrary. The first integral we obtain this time, is given by qTVq + qTVq + Zba(qfq, + fqz) + Zbp(qiq, ++q:) = constant.
(69)
Note that equation (69) is a linear expression in the arbitrary constants a and /I. Setting a # 0, and /?= 0, and in turn a = 0, and fl# 0; equation (69) splits into two independent first integrals each one corresponding to the coefficients of a and #I, namely 4:+q:+4:+4:3_2b(q:q,+~q,q:)=c,
(70)
~142+qlq2+b(qfql+3q:)=cZ.
(71)
and
A direct construction of first integrals for certain non-linear dynamical systems
143
Equation (70) reproduces the energy integral, but (71) leads to a new first integral, independent of the Hamiltonian. Leach [31] has recently discussed the existence of first integrals independent of H for general two-dimensional Hamiltonians with quadratic and cubic potentials, of which (46) is an example. He used a quite complicated series-expansion approach, which at the level of each order yields consistency conditions on the coefficients of the cubic terms. From the first few steps in this expansion he obtained necessary conditions for these coefficients in order for such an integral to exist. Our condition (67) confirms his results, since his necessary relations reduce to (67) when restricted to Hamiltonians of the type given by (46).
6. CONCLUSIONS
As indicated in the introduction, group-theoretical methods are the most powerful ones that can be used in the search for constants of the motion. That does not imply, however, that more ud hoc approaches cannot be valuable to solve certain practical problems. Even if such an ad hoc approach gives significant results, it may be considered redundant if it leads to procedures that are more cumbersome than the ones involved in the grouptheoretical approach. Instead, it is preferable to attack the problem starting from a constructive viewpoint, directly at the level of the equations of motion, appealing only to a good dose of common sense. The reader may have noticed that the method we have proposed here, is more difficult to explain than it is to understand. An additional advantage of this direct and constructive approach is that the limitations of the method are clear. It is applicable to non-linear second order equations, which are linear in the derivatives. For these systems it leads to the construction of possible energy-like first integrals, which are at most quadratic in the velocities. Note, however, that this can cover many practical applications for which one is interested in constructing a first integral, that for instance might be useful as a Lyapunov function for the study of stability properties. In the one-dimensional examples we have treated, we considered only one non-linear term in q, but with minor modifications, one can also deal with more than one polynomial term in q, as has in fact been illustrated in the two-dimensional example of Section 5. In that particular example, we were able to identify a cubic Hamiltonian, for which a simple first integral exists, independent of the energy-integral. Finally, we would like to recall the special insight gained in the broader possibilities for linear equations (Section 4). We are currently investlgating by this method; general, ndimensional, time-dependent linear systems, with a special interest in the possible connections between the existence of a first integral, and the derivability of the Newtonian equations of motion from a variational principle, a problem that is receiving much attention in theoretical physics, and which has been extensively dealt with in a recent monograph [32] by Santilli. The reader interested in recent contributions to the inverse problem in Lagrangian dynamics can consult [33-351. Acknowledyemenr-This work was completed while one of the authors (W.S.) was visiting the Department of Mechanical Engineering and Mechanics at Drexel University during the Spring term of 1979. He wishesto express his appreciation to Dr. Harry G. Kwatny, Professor of Systems Engineering in that department, for his hospitality, as well as for numerous technical discussions.
REFERENCES I. J. R. Cary, Time dependent canonical transformations and the symmetry-equals-invariant theorem. J. Math. Phvs. 18.2432-2435 (1977). 2. D.‘S. Djukic, A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians. Inr. J. Non-Linear Mech. 8,479-488 (1973). 3. G. H. Katzin and J. Levine, A gauge invariant formulation of time-dependent dynamical symmetry mappings and associated constants of motion for Lagrangian particle mechanics. I. J. Math. Phys. 17,1345-13X)(1976). 4. G. H. Katzin, J. Levine and R. N. Sane, Time-dependent dynamical symmetry mappings and associated constants of motion for classical particle systems. II. J. Math. Phys. 18.424-431 (1977). 5. G. H. Katzin and J. Levine, Time-dependent dynamical symmetries and constants of motion. III. Timedependent harmonic oscillator. J. Math. Phys. 18, 1267-1274 (1977). NLM
13.2-F
W.
144
SAKLET
and L. Y. BAHAK
6. J. D. Logan, invarianf variational principles. Academic Press, New York, NY (1977). 7. R. Abraham and J. E. Marsden. Four&ions o/ Mecllarrics (2nd edition). Benjamm/Cummmgs, Reading, Massachusetts (1978). 8. 8. Vujanovic, A‘grou&ariational procedure for finding first integrals ofdynamical systems. Int. J. Nott-Linear Me&. 5,269-278 (1970). 9. B. Vujanovic, Conservation laws ofdynamical systems via d’Alembert’s princtple. Inr. J. Non-Lineur Me& 13. 185%197(1978). 10. M. Lutzky, Symmetry groupsand conserved quantities for the harmonicoscillator. J. P/I~s. A: Marh. Gen 11. 249-258 (1978). ii. M. Lutzky, Noether’s theorem and the time-dependent harmonic oscillator. Plzvs. Leti. 68A. 34 (1978). 12. G. W. Blurnan and J. D. Cole.~iffzi~u~ir~~el~Io~s~or ~i~~erenlia~Eauutiotrs. Sminner-Verlae New York t 19741 13. C. E. Wulfman and 8. G. Wybourne, the Lie g&up di Newton’s &d LagraAge’csequatiois for the harmonic oscillator. .I. f%.vs. A: Math. Gen. 9, 507-518 (1976). 14. C. J. Eliezer. The symmetries and first integrals of some differential equations of dynamics. Hadronic J. 2, 1067-1109 (1979). 15. M. R. M. Crespoda Silva. A transformation approach for ~ndjngfirst integrals ofmotion ofdynamical systems. int. .I. Non-Linear Med. 9.241-250 (1974). 16. Dj. S. Djukic, Conservation laws in classical mechanics for auasi-coordmates. Arch. Ror. Mcch. Anal~is 56. 79-98 (l-974). 17. Dj. S. Djukic, A new first integral corresponding to Lyapunov’s function for a pendulum of variable length. ZAMP 25.532-535 (1974). 18. C. Risito, On Lyapunov stability of a system with known first integrals. Meccanica 2, 197-200 (1967). 19. H. R. Lewis Jr., Class of exact invariants for classical and quantum time-dependent harmomc oscillators. J. Math. Phys. 9. 1976-1986 (1968). 20. G. Contopoulos, On the existence of a third integral of motion. Asrronom. .!. 68, l-14 (1963). 21. S. E. Jones and W. F. Ames, Similarity variables and first integrals for ordinary differential equations. fni. J. Non-Linear Me&. 2,257-260 (1967). 22. H. R. Lewis Jr. and W. B. Riesenfeid, An exact quantum theory of the time-dependent harmonic oscillator and a charged particle in a time-dependent electromagnetic field. J. Math. Phys. 10, 1458-1473 (1969). 23. P. G. L. Leach, On the theory of time-dependent linear canonical transformations as applied to Hamiltonians of the harmonic oscillator type. J. Math. Phq’s. 18, 1608-1611 (1977). 24. P. G. L. Leach, invariantsand wavefunctions for some time-dependent harmonicoscillator-type Hamiltonians. J. Math. Phys. 18, 1902-1907 (1977). 25. P. G. L. Leach, Quadratic Namiitonians, quadratic invariants and the symmetry group SU(n). J. Marh. Phvs. 19,44@5 I (1978). 26. I(. R. Symon, Theadiabatic invariant of the linear or nonlinear oscillator. f. Math. Phys. 11,132~1330( 1970). 27. W. Sariet, Exact invariants for time-dependent Hamiltonian systems with one degree-of-freedom. J. PhJs. A : Math. Gen. 11,843-854 (1978). 28. P. G. L. Leach, Towards an invariant for the time-dependent anharmonic oscillator. J. Math. Phys. 20, 96-100 (1979). 29. G. E. Prince and C. J. Eiiezer, Symmetries of the time-de~ndent N-dimensional oscillator. J. Phq‘s. A 13, 815-823 (1980). numerical experiments. 30. M. H&non and C. Heiles, The Applicability of the third integral of motion-some Astronom
J. 69, 73-79
(1964).
31. P. G. L. Leach, A note on the H&non-Heiies problem. J. Math. Phys. 2k38-43 (1980). Me~anics I, The ~~uerseprob~em in Newfo~an Mechanics. Springer32. R. M. Santiili, Foundationsojrheorericnl Verlag New York (1978). 33. L. Y. Bahar, H. G. Kwatny and F. M. Massimo. Direct and inverse variational characterization of linear nonconservative systems. Hudronic J. 1.976-1011 (1978). 34. L. Y. Bahar and H. G. Kwatny, Exact Lagrangians for linear nonconservative systems. Hadronic J. 2,238-260 (1979). 35. W. Sariet, On the transition between second-order and first-order system< within the context of the inverse problem of Newtonian mechanics. Hadronic J. 2.407-432 (1979).
APPENDIX Comparison
with Noether
approach
Equation (17) is the Euler-Lagrange
equation corresponding to the Lagrangian. L=
+zLq C
-+I]exp (,’
fl(r.)dt’).
(A.1)
In order that the infinitesimal transformation 6=9+Etiq. T=
il
t + &r(q,t).
(A.21
determine a Noether-symmetry of the variational principle without gauge-variance, it is necessary and sufficient. following the development in [63, that the identity (A.3)
A direct construction
of first integrals for certain non-linear
dynamical
145
systems
hold for all I. q and 4. Substituting the Lagrangian given by equation (A.]) into (A.3) and rewriting pm into cvidcncc the various powers of 4 in descending order. thcrc results
I
a?
-I&4’+
(
“3’ ;; -+--iar
la,>
.* y). q + x
q-
]
[ (a/?+.)r+a$
s-uA+-0.
the result to
tA.4)
Equating the coefficients of the 4’ and the tj terms to zero in equation (A.4) immediately leads to the fact that r=?(1), and 6 = t(q). which makes the expression for q in (A.2) a function of q alone, and the one forTa function only of r. In view of these results, the vanishing coefficient of 4’ in equation (A.4) can be rewritten in the form d< _=_ dq
1 dr __ 2 ( dr
Br >
(
where the left-hand side of equation (AS) is a function of q alone, while the right-hand Hence, both sides are equal to a constant, say, a. Thus, we get
side depends only on t.
t=aq+b
(A4
(A.7) where a, b and c are constants. Finally, setting the term independent upon substitution of (A.6) to the result b=O, as well as the relation [aS+&]r+ai+(m+ The value or 4 from equation
upon incorporating
l)aa=O.
equation
(A.8) to yield, in terms of ‘I
3)ua=O.
(A.7) into (A.9) and multiplying
(A.4) leads,
tA.8)
(AS) can now be inserted into equation (2ap+ci)r+(m+
Finally,
of 4 equal to zero in equation
(A.9) by a- ‘, there results (A.10)
(2P+a-‘ir)exp(IIP(r’)d~)[c+ZaJIexp(-llP(r”)dt”)dr’]=constant.
This again gives us a condition to be satisfied by the given functions a and 1, in order that equation (17) yield a first integral through Noether’s theorem. Now, going back to our direct approach in Section 3, we see that replacing a -2”m+3) in equation (24) by its expression obtained from equation (25). gives a relation identical to (A.10). with c,, playing the role of 2a. Moreover, the invariant following from Noether’s theorem coincides with (26). This shows that both methods are essentially equivalent for the type of equations under consideration. As has been mentioned in Section 3, it can be shown !hat the extension of gauge-variance in the Noether approach, precisely corresponds to the extension (27).
Resume
:
h presente une methode directe et constructive pour trouver des integrales premieres d’une certaine classe d’equations differentielles ordinaires de seconde ordre et du type non-lineaire. L’idee generale est inspiree sur la maniere de construire l’integrale d’energie du systeme conservatif correspondant. Pour la classe d’equations etudiee, la methode developee ici, quoique elementaire. donne les memes resultats que des methodes plus avancees comne l’emploi de symmetries dans le contexte du theoreme de cas
Noether. special
Des aspects d’equations
interessants lineaires.
se Enfin,
revelent la
pour
le
methodologie
developee pour des equations scalaires est generalisee. afin d’etre applicable a des systemes d’equations. Un example de dimension deux est traite, dans lequel on montre qu’un Hamiltonien particulier du type de Contopoulos admet une integrale premiere independante de l’integrale d’energie.
Zusamnenfassung: Ein direktes aufbauendes Verfahren zur Auffindung der ersten integrate bestirenter. nichtl inearer, gewijhnl icher Differenzialgleichungen zwei ter Drdnung wi rd vorgestel It. Den Anstoss zu diesem Verfahren gab die Konstruktion des Energieintegrals fiir die Bewegungsgleichungen der zugeharigen konservat i ven Sys teme. Obwohl die Kethode fiir die hier behandelte Klasse von Clelchungen elementar ist, erzielt sie dennoch dieselben Ergebnisse wie die fortgeschritteneren
146
W. SARLET and L. Y. BAHAR
gruppentheoretischen Verfahren, wie z.B. die Verwendung van Synnnetrien im Zusammenhang mit dem Noetherschen Theorem. Die Methode zeigt einige interesante Einzelheiten, wenn sie auf den Fall 1inearer Gleichungen spezialisiert wird. Ein zweidimensionales Beispiel wird behandelt, wobei das f;r skalare Gleichuogen hergetei tete Verfahren auf deren vektoriei le Gegensticke ausgewei tet wird. Es wird gezeigt, dass als Folge fiir einen bestimmten Hamiltonischen Ausdruck Contopoulasscher Art ain erstes Integral existiert, das von dem Energieintegral unabh%gig ist.
OF FIRST INTEGRALS DYNAMICAL SYSTEMS
w. SARLET Instituut voor Theoretische Mechanica Rijksuniversiteit Gent, Krijgshtan 27169, B-9000 Gent, Belgium and L. Y. BAHARt Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, U.S.A. (Received S October 1979) Abstract-A direct, constructive approach to the problem of finding first integrals of certain nonlinear, second order ordinary differential equations is presented. The idea is motivated by the construction of the energy integral for the equations of motion of the corresponding conservative systems. Although the method developed for the class of equations studied herein is elementary, it yields the same results as the more advanced group-theoretical methods, such as the use of symmetries in the context of Noether’s theorem. The approach reveals some interesting features when it is specialized to the case of linear equations. Finally, a tw~dimensional example is considered by extending the methodology developed for scalar equations to their vector counterparts. It is shown that, as a consequence. a first integral which is independent of the energy integral exists for a particular HamiItonian of the Contopoulos type.
1. INTRGDUCT~ON
The importance of finding first integrals for ordinary differential equations has been recap nized since the earliest developments in this subject. A complete set of first integrals determines the solution of a system of differential equations. It is even more important to find a first integral in cases where it is difficult, if not impossible, to find explicit solutions, since the determination of an integral reduces the order of the differential equation, and often gives qualitative information about the behavior of the underlying dynamical system. Despite long-standing interest and research, there stil1 appear many papers emphasi~ng various approaches to derive first integrals for those cases that are not amenable to a straightforward treatment. These cases consist for example of dynamical systems governed by nonconservative forces, that give rise for the most part, to non-linear differential equations with time-dependent coefficients. The most powerful methodology by which this problem can be approached is based on group-theoretica considerations, which relate first integrals to the symmetries of the system. Moreover, a knowledge of the symmetry is often as valuable as knowing the resulting invariant. There are two closely related approaches in these methods. The first is based on the theory of the symmetries of Hamiltonian systems, generated by one-parameter families of infinitesimal canonical transformations, a recent treatment of which can be found in fl]. The other is the Noether theorem and its various generalizations. Among the many papers that have been published on this subject, we cite a few of the more recent ones [2-53, as well as the text by Logan [6]. These ideas have found their highest degree of generalization with the introduction of the concept of momentum mapping in sympiectic geometry which is described in detail in [7]. The application of Noether-type methods to practical problems require the solution of a system of partial differential equations, known as the Killing equations, or some general*Research supported by a Nato grant. tResearch supported by the U.S. Department of Energy under contracts ET-78-C-01-2092, 3088. 133
and ET-78-S-01-
134
W. S.wmand
L. Y. BAHAR
ization of those equations. Applications to various examples of the Killing equations are to be found in [2, S-11]_ In order to apply the above methods one has to know a priori a Lagrangian or ~amiltonian describing the system, or alternatively a more general variational principle such as the one described in [9]. The general theory of Lie symmetry groups, however, can be applied directly to any differential equation as indicated in [12-141. As is well known, this procedure enables one to determine the symmetries of the equation, which are not always necessarily related to first integrals [lo]. Some authors have tried to avoid the sophistication of group-theoretical methods, and have developed various ad hoc approaches for the ~nst~ction of first integrals. Al~ough such approaches are less general than group-theoretical ones, they can lead to the desired result more directly for specific problems, thus bypassing the necessity for solving simultaneous partial differential equations altogether. An example of such a method is the transformation approach proposed by Crespo da Silva [15-J. The idea behind this method is to search for a canonical transformation which reduces a non-conservative Hamiltonian to the product of a function of time, and a function of phase-space variables only, This latter function produces a constant of the motion. The idea developed in [15] is, up to this point, quite general. However, it is readily seen that it is an ad hoc approach, because only linear canonical transformations are considered, and that no information is available about the type of Hamiltonians which can be cast into the desired form by such transformations. Another area in which the generation of first integrals becomes necessary is the stability analysis of dynamical systems through the use of the second method of Lyapunov. As representative contri&tions to this field one may cite the recent work due to Djukic [16, 173, and Risito [lg]. It is, in part. to generalize existing studies to encompass the stability of linear non-conservative dynamical systems that the present work was undertaken. The purpose of this paper is to present a new direct approach to the instruction of a first integral for second order Newtonian systems. The problem is approached directly at the level of the equation of motion as given, without the need of going through a Lagrangian or Hamiltonian reformulation. Roughly speaking, the method consists of trying to construct a first integral in a manner similar to the one used in obtaining the energy integral for conservative systems, namely that of multiplying the equation of motion by an appropriate integrating factor. The present study covers all the results obtained in [15], and in addition, the constructive procedure permits the identification of the class of equations that can be treated by the method developed here. It is shown that for this class of equations the results are in agreement with those obtained from a direct application of Noether’s theorem. Furthermore, one gains new insight about the peculiarities of the linear case, which explains why the invariant of the time-dependent harmonic oscillator, found by Lewis [19] cannot be expected to be generalized to many non-linear cases. Finally, the ideas behind the present method are extended to systems with multiple degrees of freedom, in which an example of Hamiltonians studied by Contopoulos [20] is treated. The results reveal that, for a special case, a constant of the motion, which differs from the Hamiltonian, can be exhibited in closed-form. Finally, possible extensions, which are currently under study, are indicated.
2. INVARIANT
FOR
THE
EMDEN
EQUATION
Consider the equation
(1) which is known in the literature as the Emden equation [6, pp. 52-541. Following Jones and Ames [21], we eliminate the middle term of equation(l), by introducing the transformation 4 = U(r)&
(2)
A direct construction
of first integrals for certain non-linear
dynamical systems
135
which gives rise to the equation Ui+2(ti+q)i+(ti+;o)*+“Jx~=0.
(3)
Clearly, the choice u= l/t eliminates the middle term of equation (3) and simplifies it into X5 it+-=o, t4
(4)
which is the reduced form of the Emden equation. We now attempt to integrate equation (4) by multiplying it with h(t)& following the usual method employed for conservative systems, as a non-conservative term has already been eliminated, but allowing in addition the possibility of a time-dependent integrating factor. In the resulting equation, &)x2+?
x5x=0,
both terms can be written as a total time-derivative the time-dependent coefficient. We get, ;
[+h(t){i2++t-4X6}]
-$h(t)k2 -
(5)
with a correction for the derivative of
$ [h(t)~-~] f
=o.
The last two terms cannot be combined to a total time-derivative as they stand. We therefore appeal to the differential equation for replacing one of these terms. Multiplying equation (4) by x, we have x6 = _ tbxji,
(7)
which can be used in the last term of equation (6) to yield,
It becomes clear that we now have the possibility of writing the last three terms in (8) as the time-derivative of a function likef(t)xx, namely as g
[f(t)xi]
=j(t)xk +f(t)[i2 + xi].
Identification of the right hand side of equation (9) with the corresponding terms in equation (8) leads to the relations, f(c)= -$(t), (lOa) j(t)=O,
(lob)
f(t)=~[ri(t)-4t-‘h(t)],
(1Oc)
which clearly admit the solution h(t) = t,
f(t)= -4.
The above choice enables us to write the first integral of equation (4) in the form X6 tX2 + 3t3
-xi
=
constant,
(11)
which is the conservation relation derived by Jones and Ames [21] using similarity variables. It is seen that the ad hoc nature of the method resides in the fact that one must find two functions satisfying three relations, equation (lo), which is not always possible. Looking carefully at the structure of the terms produced in the steps leading to equations (6) and (9) it is also seen that the method can be extended to more general cases. As a matter of fact, it can be applied directly to the Emden equation in its original form (1). Proceeding in the same manner we obtain (12)
136
W. SARLETand L. Y. BAHAR
which we rewrite first in the form, $ [~h(c){~~+~~}]+[2t-~h(t)-~~(t)]~~-~~(t)q~=o,
(13)
and subsequently, using the differential equation to replace the last term, as $ [-/z(t){Cj*+~q6}]+ [2t-‘h(t)-~~(c)]q* Again, we can attempt to find a functionf(t) become equal to $
-+&t)[q~+2PqCj]=O.
(14)
such that the last three terms in equation (14)
CCrhmk4l =md +_m[4*+ Ml.
(15)
This gives rise to three equations for the unknown functions h(t),f(t), having as a solution h(t) = t3,
f(t)=fP.
So we obtain the first integral
t3(Q2+$I’) + t*qcj = constant,
(16)
which of course also follows from (11) and (2). We have treated this example in great detail, in order to recognize the essential features which make the approach succeed. Important is the linearity of the equation in the derivatives, while the non-linearity in q must be of polynomial character, and remain so after integration. The coefficients can be general functions of c, but they will have to satisfy a certain compatibility relation. This is clear from the fact that at the final stage, we obtained a system of equations like (lo), which in general will be overdetermined.
3. FIRST
INTEGRAL
FOR
A CLASS
OF
NON-LINEAR
EQUATIONS
From the considerations of Section 2, we conclude that a first integral will exist for the following class of equations, fj + /I(t)4 + a(t)q” = 0,
(17)
m#-1,
under certain restrictions on u and /I. We will derive these restrictions by the direct approach to the construction integral. Multiplying (17) by 2Y(t)& where y is yet undetermined, we get
of a first
2yqi + 2yb4* + 2yaqmQ = 0, which by manipulating the first and last term can be written as g
(
ul*+&
wf+’
+ 4*(2YB-Y) - --&
>
-$ (ya)q”” =o.
(18)
Making use of the differential equation for the last term, this yields
$
(
r4*+&
wT+’
>
+4’(zrS-t)+--4$ a-l z(yaXqij+Bqq)=O,
(19)
so we will find an invariant, if the remaining terms can be combined into ; (a(t)q~)=a(ri*+qij)+ciqq.
(20)
Identifying the coefficients of q4,4*, and q& we obtain the system of equations a=2y/3-Y, 2
a=--a-l m+l
(21a) -$ya),
(21b)
A direct construction
offirst integrals
for certain non-linear dynamical systems
137
(214
From (21a) and (21b), we get
m+3.
m+l’+m+l
2
y[a-‘dr-(m+
l)B]=O,
which for m # - 3 can be integrated to y=cta -2”mi3)exp(Z~IIB(I’)d~),
(22)
c, being a constant. On the other hand, from (21b) and (21~) we easily obtain a=c2 exp(,’
j?(i.)dt.).
(23)
Finally, using (22) and (23) in (21a), we get a-2/(“+3)(2/?+a-1ci) exp (2 where c3 is a constant, in fact c3 =
I’B(r)df.)=c3.
(m+3)C,
2c,
*
Equation (24) is a sufficient condition to be satisfied by the given coefficients a and fi, in order to yield a first integral. Making the substitution y(t)=a-2/(m+3),
it can be further integrated to a-2jtm+3)exp(
- --$-3 s’
jI(r)dr’)-e,
r exp (-I’
j?(r~)dr~)dr’=c,
(25)
where cq and c are arbitrary constants, where cg= -c&z,. When a and fl satisfy (25), we can compute y and a, and obtain a first integral from (19). We can choose ci = 1, ct = -c,; then the first integral is given by
42+m;l _
fl+l
a-2/(m+3)
exp(2%1)/3(r’)dr’)-c,q~exp(f jI(r’)dt.)=constant,
>
which, making use of (25), can equivalently be written as, (d2+s
q’““)
exp (2
s’fl(t.)dt')[c+c,
r exp (-I” -c&
/?(f.)df.)dt’]
exp (,‘B(r)d~‘)=constant.
(26)
Note that the present straightforward approach has given more general results than the ones obtained in [ 151. Indeed, the results mentioned in [ 153 correspond to the special case c = 0 in equations (25) and (26). Now, one might wonder how our results compare to those obtainable by the Noether approach. Let us first remark, however, that there is an additional degree of freedom in our direct method. Indeed, in the preceding calculations we tried to match the remaining terms of equation (19) by the derivative of a term like a(r)qd, as in equation (20). But, in a more general way, we could have considered a function of the form a(t)qci + b2,
(27)
with k in general constant. In the appendix we show that the results (25,26) are equivalent to those obtained by the gauge-invariant Noether approach, while it is easy to complete
138
W. SARLET and L. Y. BAHAR
the picture by showing that allowing for gauge-variance in the Noether approach, precisely corresponds to the extension (27). We now make the following two observations. Remark 1
For completeness, apart from the a priori excluded case m = - 1, we have to treat the case m= -3 separately. In this instance, equations (21a, b) immediately give rise to a condition relating a and /I, namely ci-2pa = 0,
which integrates into a = cl exp ( 2 s’ B(t’) dr’),
while the solution for a(t) and v(t) is obtained as, a=c2 exp (,’ j?(rf)dr’), y=exp(z
rfl(t’)dtj[c+c,
~exp(_,“jI(tP)dt*)dt’],
and the expression for a first integral then immediately follows from (19) and (20),
1
’ [a(t’)]-1’2 dt’ +c,a”2qo=constant,
(d2
(28)
with c1 = c2c:12. Remark 2
Let us return now to the sufficient condition (25), which ensures the existence of a first integral for equation (17). In the special case where the system is linear, and contains no damping term, i.e. /I=O, m= 1, we obtain the following condition on a(r), a(t)=(c+c,t)-2.
(29)
It is not difficult to find examples of functions a(t) which are not of the form (29), and yet allow the construction of an invariant. One such example is provided by the equation
for which an invariant is given by, as worked out in [14,19-J, t2d2 -2tqq+(tm2+
l)q2=constant.
(31)
This would seem to indicate, that the present ad hoc approach has shortcomings. As we will explain in the next section, however, the linear case, and only the linear case, allows for an interesting additional freedom in the method, which will result in the fact that in principle, an invariant can always be constructed.
4. THE
CASE
OF
A LINEAR
EQUATION
For simplicity, we treat here only the time-dependent by the equation ~+CI?(t)q=O,
harmonic oscillator, governed (32)
but a time-dependent, linear damping term can be included without significant complications. We start our attempt for the direct construction of an invariant in the usual way by multiplying (32) with 2y(t)& 2y4q + 2yw%jq = 0.
(33)
A direct construction
of first integrals for certain non-linear
dynamical systems
139
As before, we cast the first term of (33) into the form
If we follow literally the previous procedure, we would also cast the last term into the form
and eliminate the q2-term by using the differential equation. This usually leaves us with terms in qij, q(i and d2, which generically come from the timederivative of a term in 44. But in this linear case, the last term of (33) already has the structure of one of these remaining terms, which means that we need not absorb it entirely into a total time-derivative as in (35). In turn, this implies that we could have taken any function oft as the coefficient of q2 in producing terms like (35). In other words, we can simply add zero to the equation, by adding and subtracting equivalent terms. Hence we rewrite equation (33) in the form,
and make use of the differential equation to eliminate the last term, (36) Now again, we try to combine the last three terms of (36) into the form (37) which gives rise to the identifications, 4= -$9
(38a)
+&O-2,
(38b)
I$= 2(yw2 -A).
(38c)
We see that the ‘artifice of adding zero’, which only applies for the linear case, has produced this time a system of three equations with three unknowns (y, J, 4), and hence in principle is always solvable. Note that an alternative way of looking at the peculiarities of the linear case, consists in saying that for m= 1 in equation (17), k can be allowed to depend on t in the extension (27). From (38a, b) we get, 1= -&j,
(39)
while eliminating $J between (38a) and (38~) gives, rz=$+yo? A third-order
(40)
equation in y now follows from elimination of L between (39), and (40), $7 + 205 + 2ywci,= 0.
(41)
This equation coincides with the one obtained by Lutz&y in [ll]. It can be reduced to a second-order equation in the following way. First, we make the substitution Y’P2,
(42)
which allows rewriting equation (41) in the form p $ (li+w2p)+3p(&5+02p)=o.
(43)
Equation (43) has the integrating factor p2, from which it easily follows that p must satisfy
W. SARLETand L. Y. BAHAR
140
the second-order
differential equation cp-3,
b+o2p=
C constant.
(44)
An invariant can now be calculated from (36) and (37). Moreover, the functions y, 1 and 4 can all be expressed in terms of the single function p using (42) and (38). In view of (44), this invariant finally takes the form, Cp-2q2+(pq-~q)2=constant.
(45)
For C= 1, equation (44) and (45) are precisely the results obtained by Lewis [19]. These results have been shown to be important, for instance, for a quantum-mechanical treatment ofthe time-dependent oscillator [22]. They have also been generalized to multi-dimensional but linear systems by Leach [23-25-j, by making use of linear canonical transformations. Other studies aimed at generalizing the Lewis-invariant to non-linear equations have been performed by Symon [26], Sarlet [27], and Leach [28]. For additional aspects, [14] and [29] can also be consulted. Attempts for finding similar results for non-linear equations have not been very successful as can be seen from the complicated expansion procedures in [28]. From the present heuristic approach, we have learned, that the linearity of the equation plays a crucial role for deriving the Lewis-invariant, and that for non-linear problems, the ultimate system of differential equations becomes overdetermined, which means that results will only be found for specific choices of the time-dependent coefficients.
5. A MULTIDIMENSIONAL
NON-LINEAR
EXAMPLE
Consider the dynamical system described by the Hamiltonian (46) b and c being constants. It corresponds to the coupled system of non-linear second-order equations, ci’i+q,+2bq,qz=O, d2fq2+bq:+3cq;=0,
(47)
ij+q+Q(qh=O,
(48)
which we write in matrix form as
where q = col(q,, q2) and (49) Hamiltonians of type (46) have been discussed by Contopoulos [20], and later on by many others, with regard to the existence of a first integral independent from the Hamiltonian. For the special choice b= 1, c= -l/3, (46) gives the Hamiltonian introduced by Htnon and Heiles [30]. In the spirit of the constructive approach we have used to find a first integral for a scalar equation, we start by multiplying the vector-equation (48) on the left by 20%), where V is a possibly time-dependent pose. We get,
v=v’,
2 by 2 matrix, and the superscript T denotes the trans-
24TV4 + 2cfvq + 2ijTw2(q)q = 0, or (50) In this last reformulation,
we have used the insight gained in the previous section, that is,
A direct construction
of first integrals
for certain
non-linear
dynamical
141
systems
we have added zero to the equation through the terms involving W, which is a possibly time-dependent symmetric matrix. The reason for this procedure, is due to the presence of the linear term in q in the given equation (48). There is no loss in generality in assuming W to be symmetric, since qTAq identically vanishes when A is skew-symmetric. We put, V(t) =
40 PO) D(t)- r(t) ’
(
>
(51)
then the last term of equation (50), when computed explicitly gives, 2QTVM)q=4@q,q,9,
+2ybq:q’,+6Bc&,
+4Pq,q,&+2Bbq:&
+%&j,.
(52)
Following the procedure outlined for scalar equations, we normally would combine these terms into the derivative of some polynomial in q with time-dependent coefficients, then make a correction for the derivatives of these coefficients, and finally replace these correction terms by making use of the differential equation. The important point is that these correction terms need to be of polynomial character in q, that is, free of derivatives. This is always automatically the case for a single equation, but it is clear that for multidimensional systems, it can impose additional restrictions, which are essentially due to the coupling of the equations. Roughly speaking, we need to require the terms (52) to have the structure of an exact differential in q, and q2, but not necessarily with the correct coefficients. Inspection of (52) shows that, for the case at hand, we need the requirements, ab= yb,
(53)
/3c=$b.
(54)
Indeed, under these assumptions, we can write 24’VfJ(q)q=$
(2abq:q,+Zflbq$q,
+$3bq:
+2ycq:)-2cibqfq, -2Sbq:q,
-$jbq:
-2vcq;,
(55)
and the equation of motion can be invoked to replace the four correction terms. For b = 0, the equations in q1 and q2 are uncoupled, and one easily finds two constants of the motion, corresponding to the two uncoupled parts of the Hamiltonian. we further need only to investigate the case b#O. Then the correction terms in (55) can be written as -qTr(t)r2(q)q,
with r(t)=
f-2_
ls
and can be replaced from the equation of motion (48) by, - qTrR(q)q = $I-4 + qTrq. So, going back to equation (50), and taking into account the manipulations on the last term, we get
$ (tjTVtj + qTWq + 2abqfq, + Vbqiq,
+ $bq:
we performed
+ 2vcqz)
-qTVq+2tjT(v-w)q-qT(vL-r)q+q’rij=o.
(57)
In this manner we reach the final step in the approach, which is that in order to find a first integral the remaining non-derivative terms in (57) should come from, $ 6jTD0)B)= cjTD(t)Q+ qTD(t)ii + qTD(t)Q, D being an arbitrary time-dependent equations (57) and (58) we get,
matrix. Identifying
the corresponding
(58)
terms in
D(t) = r(t),
(59a)
D(t)= -%‘(t)+A,(t),
(59b)
142
W. SARLETand L. Y. BAHAR
w =Wt)
(59d
- WO),
I-(t)= Ii’(t) + A*(t),
(59d)
where A,(t), and A&) are arbitrary skew-symmetric matrices. From (59a, b, d) we get Ki’+A,= -%‘+A,, and, recalling that V and W are symmetric, we conclude that W(r)= -V(r),
60)
A2(r)=Al(r), which we will denote by A(r).
(61)
Furthermore, equation (59~) shows that fi must be symmetric, which in view of (59b) implies that A should be constant. Now, from (53) and (54), and in view of the fact that b #O, we get a = y, and either /?=O or c= b/3. Hence, two cases have to be considered separately. Case 1 a=y,
p=o.
(62)
We then have
“=(1;I), r$
;).
(63)
Since from (59a, b) the symmetric part of r should equal -V, we conclude that a must be constant, and therefore we have, &D=A,=A,=V=W=(),
(64)
while from (59c), we get v=w.
(65)
From (57) and (58), we conclude that a first integral is given by qTVq + qTVq + 2a(bqtq, + cqz)= constant,
(66)
which in view of the nature of V is nothing else but a multiple of the energy integral, and therefore does not produce a new result. Case 2 a=y,
c = b/3.
(67)
We now have
v=
(a p) B a’
1-+.
For the same reason as before, we conclude that r must be zero, which implies (64) and (65), and of course that a and p are constant, but arbitrary. The first integral we obtain this time, is given by qTVq + qTVq + Zba(qfq, + fqz) + Zbp(qiq, ++q:) = constant.
(69)
Note that equation (69) is a linear expression in the arbitrary constants a and /I. Setting a # 0, and /?= 0, and in turn a = 0, and fl# 0; equation (69) splits into two independent first integrals each one corresponding to the coefficients of a and #I, namely 4:+q:+4:+4:3_2b(q:q,+~q,q:)=c,
(70)
~142+qlq2+b(qfql+3q:)=cZ.
(71)
and
A direct construction of first integrals for certain non-linear dynamical systems
143
Equation (70) reproduces the energy integral, but (71) leads to a new first integral, independent of the Hamiltonian. Leach [31] has recently discussed the existence of first integrals independent of H for general two-dimensional Hamiltonians with quadratic and cubic potentials, of which (46) is an example. He used a quite complicated series-expansion approach, which at the level of each order yields consistency conditions on the coefficients of the cubic terms. From the first few steps in this expansion he obtained necessary conditions for these coefficients in order for such an integral to exist. Our condition (67) confirms his results, since his necessary relations reduce to (67) when restricted to Hamiltonians of the type given by (46).
6. CONCLUSIONS
As indicated in the introduction, group-theoretical methods are the most powerful ones that can be used in the search for constants of the motion. That does not imply, however, that more ud hoc approaches cannot be valuable to solve certain practical problems. Even if such an ad hoc approach gives significant results, it may be considered redundant if it leads to procedures that are more cumbersome than the ones involved in the grouptheoretical approach. Instead, it is preferable to attack the problem starting from a constructive viewpoint, directly at the level of the equations of motion, appealing only to a good dose of common sense. The reader may have noticed that the method we have proposed here, is more difficult to explain than it is to understand. An additional advantage of this direct and constructive approach is that the limitations of the method are clear. It is applicable to non-linear second order equations, which are linear in the derivatives. For these systems it leads to the construction of possible energy-like first integrals, which are at most quadratic in the velocities. Note, however, that this can cover many practical applications for which one is interested in constructing a first integral, that for instance might be useful as a Lyapunov function for the study of stability properties. In the one-dimensional examples we have treated, we considered only one non-linear term in q, but with minor modifications, one can also deal with more than one polynomial term in q, as has in fact been illustrated in the two-dimensional example of Section 5. In that particular example, we were able to identify a cubic Hamiltonian, for which a simple first integral exists, independent of the energy-integral. Finally, we would like to recall the special insight gained in the broader possibilities for linear equations (Section 4). We are currently investlgating by this method; general, ndimensional, time-dependent linear systems, with a special interest in the possible connections between the existence of a first integral, and the derivability of the Newtonian equations of motion from a variational principle, a problem that is receiving much attention in theoretical physics, and which has been extensively dealt with in a recent monograph [32] by Santilli. The reader interested in recent contributions to the inverse problem in Lagrangian dynamics can consult [33-351. Acknowledyemenr-This work was completed while one of the authors (W.S.) was visiting the Department of Mechanical Engineering and Mechanics at Drexel University during the Spring term of 1979. He wishesto express his appreciation to Dr. Harry G. Kwatny, Professor of Systems Engineering in that department, for his hospitality, as well as for numerous technical discussions.
REFERENCES I. J. R. Cary, Time dependent canonical transformations and the symmetry-equals-invariant theorem. J. Math. Phvs. 18.2432-2435 (1977). 2. D.‘S. Djukic, A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians. Inr. J. Non-Linear Mech. 8,479-488 (1973). 3. G. H. Katzin and J. Levine, A gauge invariant formulation of time-dependent dynamical symmetry mappings and associated constants of motion for Lagrangian particle mechanics. I. J. Math. Phys. 17,1345-13X)(1976). 4. G. H. Katzin, J. Levine and R. N. Sane, Time-dependent dynamical symmetry mappings and associated constants of motion for classical particle systems. II. J. Math. Phys. 18.424-431 (1977). 5. G. H. Katzin and J. Levine, Time-dependent dynamical symmetries and constants of motion. III. Timedependent harmonic oscillator. J. Math. Phys. 18, 1267-1274 (1977). NLM
13.2-F
W.
144
SAKLET
and L. Y. BAHAK
6. J. D. Logan, invarianf variational principles. Academic Press, New York, NY (1977). 7. R. Abraham and J. E. Marsden. Four&ions o/ Mecllarrics (2nd edition). Benjamm/Cummmgs, Reading, Massachusetts (1978). 8. 8. Vujanovic, A‘grou&ariational procedure for finding first integrals ofdynamical systems. Int. J. Nott-Linear Me&. 5,269-278 (1970). 9. B. Vujanovic, Conservation laws ofdynamical systems via d’Alembert’s princtple. Inr. J. Non-Lineur Me& 13. 185%197(1978). 10. M. Lutzky, Symmetry groupsand conserved quantities for the harmonicoscillator. J. P/I~s. A: Marh. Gen 11. 249-258 (1978). ii. M. Lutzky, Noether’s theorem and the time-dependent harmonic oscillator. Plzvs. Leti. 68A. 34 (1978). 12. G. W. Blurnan and J. D. Cole.~iffzi~u~ir~~el~Io~s~or ~i~~erenlia~Eauutiotrs. Sminner-Verlae New York t 19741 13. C. E. Wulfman and 8. G. Wybourne, the Lie g&up di Newton’s &d LagraAge’csequatiois for the harmonic oscillator. .I. f%.vs. A: Math. Gen. 9, 507-518 (1976). 14. C. J. Eliezer. The symmetries and first integrals of some differential equations of dynamics. Hadronic J. 2, 1067-1109 (1979). 15. M. R. M. Crespoda Silva. A transformation approach for ~ndjngfirst integrals ofmotion ofdynamical systems. int. .I. Non-Linear Med. 9.241-250 (1974). 16. Dj. S. Djukic, Conservation laws in classical mechanics for auasi-coordmates. Arch. Ror. Mcch. Anal~is 56. 79-98 (l-974). 17. Dj. S. Djukic, A new first integral corresponding to Lyapunov’s function for a pendulum of variable length. ZAMP 25.532-535 (1974). 18. C. Risito, On Lyapunov stability of a system with known first integrals. Meccanica 2, 197-200 (1967). 19. H. R. Lewis Jr., Class of exact invariants for classical and quantum time-dependent harmomc oscillators. J. Math. Phys. 9. 1976-1986 (1968). 20. G. Contopoulos, On the existence of a third integral of motion. Asrronom. .!. 68, l-14 (1963). 21. S. E. Jones and W. F. Ames, Similarity variables and first integrals for ordinary differential equations. fni. J. Non-Linear Me&. 2,257-260 (1967). 22. H. R. Lewis Jr. and W. B. Riesenfeid, An exact quantum theory of the time-dependent harmonic oscillator and a charged particle in a time-dependent electromagnetic field. J. Math. Phys. 10, 1458-1473 (1969). 23. P. G. L. Leach, On the theory of time-dependent linear canonical transformations as applied to Hamiltonians of the harmonic oscillator type. J. Math. Phq’s. 18, 1608-1611 (1977). 24. P. G. L. Leach, invariantsand wavefunctions for some time-dependent harmonicoscillator-type Hamiltonians. J. Math. Phys. 18, 1902-1907 (1977). 25. P. G. L. Leach, Quadratic Namiitonians, quadratic invariants and the symmetry group SU(n). J. Marh. Phvs. 19,44@5 I (1978). 26. I(. R. Symon, Theadiabatic invariant of the linear or nonlinear oscillator. f. Math. Phys. 11,132~1330( 1970). 27. W. Sariet, Exact invariants for time-dependent Hamiltonian systems with one degree-of-freedom. J. PhJs. A : Math. Gen. 11,843-854 (1978). 28. P. G. L. Leach, Towards an invariant for the time-dependent anharmonic oscillator. J. Math. Phys. 20, 96-100 (1979). 29. G. E. Prince and C. J. Eiiezer, Symmetries of the time-de~ndent N-dimensional oscillator. J. Phq‘s. A 13, 815-823 (1980). numerical experiments. 30. M. H&non and C. Heiles, The Applicability of the third integral of motion-some Astronom
J. 69, 73-79
(1964).
31. P. G. L. Leach, A note on the H&non-Heiies problem. J. Math. Phys. 2k38-43 (1980). Me~anics I, The ~~uerseprob~em in Newfo~an Mechanics. Springer32. R. M. Santiili, Foundationsojrheorericnl Verlag New York (1978). 33. L. Y. Bahar, H. G. Kwatny and F. M. Massimo. Direct and inverse variational characterization of linear nonconservative systems. Hudronic J. 1.976-1011 (1978). 34. L. Y. Bahar and H. G. Kwatny, Exact Lagrangians for linear nonconservative systems. Hadronic J. 2,238-260 (1979). 35. W. Sariet, On the transition between second-order and first-order system< within the context of the inverse problem of Newtonian mechanics. Hadronic J. 2.407-432 (1979).
APPENDIX Comparison
with Noether
approach
Equation (17) is the Euler-Lagrange
equation corresponding to the Lagrangian. L=
+zLq C
-+I]exp (,’
fl(r.)dt’).
(A.1)
In order that the infinitesimal transformation 6=9+Etiq. T=
il
t + &r(q,t).
(A.21
determine a Noether-symmetry of the variational principle without gauge-variance, it is necessary and sufficient. following the development in [63, that the identity (A.3)
A direct construction
of first integrals for certain non-linear
dynamical
145
systems
hold for all I. q and 4. Substituting the Lagrangian given by equation (A.]) into (A.3) and rewriting pm into cvidcncc the various powers of 4 in descending order. thcrc results
I
a?
-I&4’+
(
“3’ ;; -+--iar
la,>
.* y). q + x
q-
]
[ (a/?+.)r+a$
s-uA+-0.
the result to
tA.4)
Equating the coefficients of the 4’ and the tj terms to zero in equation (A.4) immediately leads to the fact that r=?(1), and 6 = t(q). which makes the expression for q in (A.2) a function of q alone, and the one forTa function only of r. In view of these results, the vanishing coefficient of 4’ in equation (A.4) can be rewritten in the form d< _=_ dq
1 dr __ 2 ( dr
Br >
(
where the left-hand side of equation (AS) is a function of q alone, while the right-hand Hence, both sides are equal to a constant, say, a. Thus, we get
side depends only on t.
t=aq+b
(A4
(A.7) where a, b and c are constants. Finally, setting the term independent upon substitution of (A.6) to the result b=O, as well as the relation [aS+&]r+ai+(m+ The value or 4 from equation
upon incorporating
l)aa=O.
equation
(A.8) to yield, in terms of ‘I
3)ua=O.
(A.7) into (A.9) and multiplying
(A.4) leads,
tA.8)
(AS) can now be inserted into equation (2ap+ci)r+(m+
Finally,
of 4 equal to zero in equation
(A.9) by a- ‘, there results (A.10)
(2P+a-‘ir)exp(IIP(r’)d~)[c+ZaJIexp(-llP(r”)dt”)dr’]=constant.
This again gives us a condition to be satisfied by the given functions a and 1, in order that equation (17) yield a first integral through Noether’s theorem. Now, going back to our direct approach in Section 3, we see that replacing a -2”m+3) in equation (24) by its expression obtained from equation (25). gives a relation identical to (A.10). with c,, playing the role of 2a. Moreover, the invariant following from Noether’s theorem coincides with (26). This shows that both methods are essentially equivalent for the type of equations under consideration. As has been mentioned in Section 3, it can be shown !hat the extension of gauge-variance in the Noether approach, precisely corresponds to the extension (27).
Resume
:
h presente une methode directe et constructive pour trouver des integrales premieres d’une certaine classe d’equations differentielles ordinaires de seconde ordre et du type non-lineaire. L’idee generale est inspiree sur la maniere de construire l’integrale d’energie du systeme conservatif correspondant. Pour la classe d’equations etudiee, la methode developee ici, quoique elementaire. donne les memes resultats que des methodes plus avancees comne l’emploi de symmetries dans le contexte du theoreme de cas
Noether. special
Des aspects d’equations
interessants lineaires.
se Enfin,
revelent la
pour
le
methodologie
developee pour des equations scalaires est generalisee. afin d’etre applicable a des systemes d’equations. Un example de dimension deux est traite, dans lequel on montre qu’un Hamiltonien particulier du type de Contopoulos admet une integrale premiere independante de l’integrale d’energie.
Zusamnenfassung: Ein direktes aufbauendes Verfahren zur Auffindung der ersten integrate bestirenter. nichtl inearer, gewijhnl icher Differenzialgleichungen zwei ter Drdnung wi rd vorgestel It. Den Anstoss zu diesem Verfahren gab die Konstruktion des Energieintegrals fiir die Bewegungsgleichungen der zugeharigen konservat i ven Sys teme. Obwohl die Kethode fiir die hier behandelte Klasse von Clelchungen elementar ist, erzielt sie dennoch dieselben Ergebnisse wie die fortgeschritteneren
146
W. SARLET and L. Y. BAHAR
gruppentheoretischen Verfahren, wie z.B. die Verwendung van Synnnetrien im Zusammenhang mit dem Noetherschen Theorem. Die Methode zeigt einige interesante Einzelheiten, wenn sie auf den Fall 1inearer Gleichungen spezialisiert wird. Ein zweidimensionales Beispiel wird behandelt, wobei das f;r skalare Gleichuogen hergetei tete Verfahren auf deren vektoriei le Gegensticke ausgewei tet wird. Es wird gezeigt, dass als Folge fiir einen bestimmten Hamiltonischen Ausdruck Contopoulasscher Art ain erstes Integral existiert, das von dem Energieintegral unabh%gig ist.