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Adiabatic invariants for dynamical systems with one degree of freedom
hr. J. Non-Linear Mechanics. Vol. 16. No. S/t, pp. 489-498, 1981 Rimed in Great Etritain
ADIABATIC INVARIANTS FOR DYNAMICAL SYSTEMS WITH ONE DEGREE OF FREEDOM DJ. S. DJUKIC~ Department of Engineering Science University of Cincinnati, Cincinnati, OH 45221. USA (Received 29 May 1979)
Abstract-Adiabatic invariants for dynamical systems with one degree of freedom. whose equation of motion is (I}, and where the existence of the corresponding Hamilton action integral is not imposed, are estabhshed. The adiabatic invariants may vary according to their structure. Using the theory a few particular problems, including non-autonomous Duffing and Van der Pol oscillators. are andysed. Finally, it is indicated how the adiabatic invariants can be used for finding approximate solutions, stability analyses and for plotting phase curves. I. INTRODUCTION
1n this paper, I propose to extend the idea of adiabatic invariants to dynamical systems with one degree of freedom, whose equation Of motion is &td(7)q=&f(z,
q, 4)
7=&f
(1)
indicates d/dt; ~, d2/dt2; ’ , d/&; and (I is the generalized coordinate; t is time, w is an arbitrary function of slowly varying time r, Eis a small parameter andfis an arbitrary function of the variables r, q and 4. By ‘adiabatic invariant’ for the dynamical system, we mean a function LI(C,7, q, 4) whose time derivative, by virtue of the equation of motion (1 X contains only terms higher than some particular order of the parameter E. If the time derivative of the function D is equal to zero then the function is an exact invariant, i.e. first integral, for the equation of motion (1). The basic results on adiabatic invariants are perhaps best stated in the work of Burgers [26]. In the last ten years, after Kruskal’s paper [27], there has been a growing interest in finding adiabatic (see [ 12-163) and exact (see [lo], [12], [13], [16], [20-231) invariants. All of the results are valid if the equation (1) can be derived from the corresponding Hamilton’s variational principle. If the dynamical system is purely non-conservative (for example the Van der Pol oscillator), that is, when the system cannot be described by H~iltonian canonical equations, the existing theory fails. Usually, adiabatic invariants are obtained by approximately satisfying the appropriate necessary conditions for the existence of the exact invariants. In fact, the adiabatic invariants form a series in powers of the parameter E. The leading term in the series is Born’s adiabatic invariant [l 11, i.e. it is energy divided by frequency. The central feature of the present discussion is the establishment of abiabatic invariants for the dynamical system (l), but where it is not assumed that the corresponding Hamiltonian canonical formalism exists. The theory is based on Noether’s theory for classical non-conservative mechanics [3] (for an extended review of the Noether’s theory consult [2-g]) and the use of Krylov-Bogoliubov-Mitropolski (KBM) asymptotic technique (see for example [l]). The adiabatic invariants obtained are series with respect to the small parameter a. Although the procedure outlined here is straightforward in principle, it is difficult to carry the procedure to a higher order without a prohibitive amount of labor. For this reason, the calculation of adiabatic invariants is concluded with the first order terms with respect to E. At the same time, it means that the time derivative Of the adiabatic invariants is proportional to &2,Here, the leading term in the adiabatic invariant series may be of various structures. Therefore, even if the dynamical system is a where
*On leave from the University of Novi Sad, 21000 Novi Sad, Yugostavia. 489
490
Dj S DJUKIC
Hamiltonian canonical system, the paper gives a new insight into the nature of the adiabatic invariant series. The general results of the paper can be easily applied to particular problems. We therefore restrict our attention to problems which are new or which illustrate particular features of the present adiabatic invariants. Typically, adiabatic invariants are used for practical purposes in quantum mechanics (see for example [ 173 and [20]). In this paper, it is indicated how adiabatic invariants may be used in vibration theory, stability analyses and for the construction of phase curves. 2
NOETHERlAN
THEORY
IN
KRYLOV-BOGOLIUBOV-WITROPOLSKI VARIABLES
To attack the problem of finding the adiabatic invariants, for the dynamical system (l), we must slightly generalize the results of the [3) and [28]. It is obvious that we may construct the Lagrangian and Hamiltonian function, for the conservative part [left-hand side of the equation (l)] of the system, in the following form
L=f[qZ -d(.r)q2];
H=gJ?+d(r)q2];
pL$j
(2)
where p is generalized momentum. Let us consider a continuous one-parameter transformation of the time, generalized coordinate, generalized velocity and generalized momentum ? 4, P)
f=:t+P&
4 “4 + I446 L 4, PI
p$f+r[i-qS+#t,
79%PJI
(3)
where p is a small parameter of the transformation and JI, 4, @ and v are functions of time, slow time, generalized coordinate and generalized momentum. Thus, corresponding to (3) there exists an infinitesimal transformation of the form At =p$, Aq =p$, A4 z/.+$-4$+0)
Ap Z/N.
(4)
Further, let us assume that the following expression V=(pd-H)
dt
(5)
is invariant in the sense that
(6)
1
AV= p$! -B(r, 4. p) df-[pi--H(t, q, p)] dt=O C when it is the object of the transformation. Combining (3), (4) and (6), developing the term B in series around the point t, q, p and retaining only members linear in the small parameter p, we arrive at the following expression
(7) which after a simple manipulation AV=p
becomes
-&#+H)+(cj-
-F/($-q$)+($+g
Now, let us consider the transformation KBM variables cxand 6, defined by q=a(t)
(8) of the Hamiltonian
cos h(t), p= -aw
sin 6.
variables q and p into the (9)
Adiahatx
InvarIants
tor dynamical
systems with one degree of freedom
491
Thus the original second-order differential equation ( 1), where 4 = p, for q may be replaced (see more details in [I]. p. 165) by the following two first-order differential equations 2 = - 2 EUJ’sin’ d - 5 f(5, r Cos 6. -20) sin d;)sin 6 flJ 0) 8=OJ - .fi 01’ sin b cos b - 2- f( 310 0
7, 2 cos 6
-
(10)
ZOJ sin 6) cos 6
for the KBM variables. where L’)‘= d&dr. At the same time, using (2) and (9) the equations (7) and (8) reduce to AV = ~1c v(r;icos 6 - ub sin b + m sin 6) -cm sin S(d + 0) L
- f$zw2 cos b - *EoJ~~~ -ZOJC$
1
a2uJ2
cos’ 6 - Ic; -
2
dt
(11)
r2u12 + icosd-aasin6 sin S-tf5 2 >(
+ 20~ sin d [Y+ (io sin b + ~oJ’&sin 6 + sod
COS S)$]
>
+(-iwsind-m)‘Esinb-rw6’cosS+aw2cos6-i$) x [(ti cos 6 - zd sin 6)$ - 4]-
(ao sin 6)@
--~f[&-_(icosd-rdsin6)$]
dt
(12)
I while the Hamiltonian
(2) becomes r
H=F.
W
(13)
Remembering that the dynamical system under consideration moves in agreement with the equations (10). and assuming that the functions @.4 and tj satisfy the algebraic equation rwdr sin d = --~f($ + ro$ sin b)
(14)
we may deduce the following theorem from (6) and (12) Theorem 1. If. under the continuous infinitesimal one-parameter transformation (4). which satisfies the algebraic equation (9). the expression (5) is invariant in the sense of equation (6). then the quantity D=rwr$
r20J2 sin 6+--2 *
(15)
is constant of motion. i.e. it is an exact invariant for the dynamical system under consideration. Now. combining (6). (10). (11) and (14) we obtain the necessary condition for the functions 4 and Ic/.that must be satisfied for the function V to be invariant under the infinitesimal transformation (4) @to’ Cos b + fjrw sin 6 + *wo’Ea2 cos2 6 x21d +-- 2 Icj-_Es(zw$ sin 6+$)=0
(16)
while the function Q, must satisfy the algebraic equation (14). Here we may remark that after integration of the equation (16) with respect to 4 and $ the function @ can be easily found from equation (14). Also, from this analysis, one can establish the next theorem. Theorem 2. If the equation ( 16) admits a solution in 4 and $, then the equation of motion (1) admits an exact invariant ( 15) in the KBM variables. The theorem is in fact a recipe for finding the exact invariants. It is important that the equations (15) and (16) define a class of
49’
DJ S DJUKK
exact invariants because 4 and $ may be any particular solution of equation (16). Here, we make one important remark. The solution of the equation (16) and the first integral (15) do not depend on the quantity v, which appears in the transformation law for the generalized momentum (3). At first it may appear that this quantity may take arbitrary values. It can be seen that this supposition is erroneous if we recall that in Hamiltonian mechanics the generalized momenta are known functions of time, generalized coordinates and generalized velocities. Hence, the transformed generalized momentum p is completely determined by the transformed quantities r, 4 and dg/dr, i.e. the quantity v can be obtained as a function of $, 4 and Q, [see (3)]. Thus the logic of Hamilton’s mechanics is maintained. 3
THE
ADIABATIC
INVARIANTS
The class of exact solutions of the equation (16), i.e. the corresponding class of exact invariants, is extremely limited by a particular choice of the functions o andfin differential equation (1). For this reason, we will concentrate our attention on the approximate solutions of the equation (16). At the same time, it means that the invariance condition for the expression V, that is, that AV = 0, will not be satisfied. Namely, we will search for such solution of the equation (16) for which the AV is proportional to s2. It will lead us to the adiabatic invariants for the dynamical system (1). Let us consider the following series $ = Jl&, w, 6, +=&(a,04
7, t) + EJ/ I(a, w, 6, T, t, 0’) + c2.
6, r, t)+i$r(a,
048, T, t,o’)+e2.
..
(17)
..,
(18)
with respect to the small parameter E, as a solution of the equation (16). Here, the eO, $1,. . * 7 $0, 41,*.. 9 are unknown functions of the variables indicated in the brackets. Substituting (17) and (18) into the equation (16), using (10) to eliminate oi and 8 from the obtained result and equating with zero the terms in corresponding powers of E, because residual AV of the equation (16) has to be proportional to e2 for every E,we have the following two equations &o cos 6+ rk-++
?$)sin~+~(C!$!?+~~)=O
34, 84, sin 6 o~+c~‘~+~+~
4,ao2 cosS+ao
(19)
a+, 84,
i
_;C!&
sin6+Tw’(l-cos26)]-Aa%
-!fcos~? ( a2m2
+kw’.sin26
%o 0’;;;;;
+~$pd(l+COS26)+~ >I
i
_ wo
1i(ljo 1 --( +w$+g+-
w&5
?r
-
kg
-fcosS+Gsin26 u
= c$J+ aw$,f
_I
fsin 6+:0’(1
1
-cos 26)
[ >I
sin 6.
(20)
Solving these equations with respect to &,,4r, $0 and $1
wehave (21)
where
T, a cos 6, -am + Sd7,
a, w, w’)
sin 6) sin 6d 6
(22)
Adiabatic invariants for dynamxal systems with one degree of freedom
493
and where S and S, are arbitrary functions of the corresponding variables. Here, we must underline the fact that the solution (21) is not a unique solution to the equations (19) and (20). Now, ~mbining (17), (18) and (21) with (1 St we have the quantity D as I) = aoS(a, w, t) + EawF(a, u, T, 6, m’)
(23)
whose time derivative by virtue of the equations of motion (10) is
- (F+ag)[f
sin S+iw’(l
-cos 26)] (24)
Therefore, the quantity D may be called an adiabatic invariant for the dynamical system whose equation of motion is given by (1) or (10). Also, using (4), (14), (17), (18) and (21) we have the infinitesimal transformation in the form
under which the function V, given by (5), has the change of the order Em.For this reason, we may say that the V is adiabatic invariant when it is the object of the transformation of variables (25) and (26). Therefore, the existence of the adiabatic invariant D is caused by the adiabatic invariance of the function V. Using (13), the leading term awS in the adiabatic invariant (23) becomes
where the function S is still arbitrary. Hence, the term may be of a general structure, which may be completely different than the energy (H) divided by the frequency @I) as it is the case in the existing literature. This gives a new insight into the nature of adiabatic invariants. Now, let us suppose that the functions S and S, are of the following form S=ka*o” 7S o--0
CW
where k, b and c are arbitrary constants. Selecting particular values for these constants and using (13), (22) and (23) we have some special types of the adiabatic invariants.
(a) Classical type For k = l/2, b = 1, c = 0 we have an adiabatic invariant DI=E
--E sfksin26
(28)
where the leading term is energy divided by frequency.
(b) Energytype Fork=1/2,b=l
andc=l D,=H-+
wehave &
H(26+sin
,/(2H) 2s) - 0
i.e. the adiabatic invariant with energy as the leading term.
f sin 6dS f
1
Dj. S. DJUKIC
494
(cl 0, S type For k = 1, b = - 1 and c = 1 we have the following adiabatic invariant D, = 0’ - 2dEb,
(30)
which is a function of the w and 6 only.
4
APPLICATIONS
The above general results can easily be applied to a particular problem. The application requires calculation of the integral f f sin 6 da, for a particular structure of J Obviously, the functionfmust be expressed in the KBM variables. It is interesting that the adiabatic invariant D,, which is given by (30), is of the same structure for every& For the reason that the adiabatic invariants are approximately constants of motion, the ‘numbers’ D1, D2, . . . can be calculated by the initial conditions. 4.1. Classical time-dependent harmonic oscillator As the first example, let us consider the classical time-dependent harmonic oscillator. In this case the differential equation of motion is given by (l), where j-=0. The corresponding
(31)
adiabatic invariants of classical (28) and energy (29) type are D 1=- H w
(
sin 26 , (ti=m’)
1-s
>
D,=H
l-
6
1.
(26 + sin 26)
[
(3.2)
(33)
While the second adiabatic invariant D2 is unknown in the literature, the first one differs from those proposed by Symon [16] for the following term H o’2&2 cos2 6. 405
The term and its time derivative are of the order .s2. Hence, the qualities of the Symon’s invariant and the D, are the same. 4.2. Pendulum of variable length The equation of motion of a simple pendulum with small angular displacement is of the form (1) with
Now, the classical type adiabatic invariant (28) is D, =z+Eg(3
sin 26-86).
In the next section of the paper, it will be indicated how the adiabatic invariants can be used in stability analyses. For this purpose, the following invariant for the same problem is much more convenient D,=a20-3+~w-Su@+sin
26).
The result is obtained by making use of (13), (22) and (23) and for S=aoe4,
where j? is constant.
3a
SO=B~~’
(36)
Adiabatic
mvariants
for dynamlcal
systems with one degree of freedom
495
4.3. Non-autonomous Dufflng equation
In this case the characteristic
functionfis f=bo(r)q3+W)
(38)
where b, and bl are arbitrary functions of the slowly varying time T. The adiabatic invariant of the classical type (28) is H W’H sin 26 - Ao2 D, =w--E~
y
H2 cos4 6 +J(2H)b,(r)
cos 6
1
(39)
4.4. Non-autonomous Van der Pol oscillator For the governing differential equation of motion 4 + w2(z)q = E[( 1 -q2)4 + b(r)] where b is an arbitrary function of
7,
(40)
the classical type adiabatic invariant is
+ 4H sin 26 - 8,/(2H)b(r)
cos 6 -
.
(41)
4.5. An oscillator with quadratic non-linear term If the characteristic function of the problem is
f =b(r)q2
(42)
where the b is an arbitrary function of r, the equation (28) yields the following adiabatic invariant --E&Hsin26+
2 2 3. b(r)H3’2 cos3 6 -JX
(43)
whose time derivative is of s2 order. For the same problem Symon [16] has obtained an adiabatic invariant, but whose time derivative is of the order E. 5.
DISCUSSION
In this paper I have proposed a procedure for finding adiabatic invariants for a dynamical system of one degree of freedom, whose equation of motion is given by (1). The results obtained are very general and valid for any structure of the ‘non-conservative force’J Although it is assumed that the system is a classical dynamical system, that is, that o and f are real, the adiabatic invariant D has the same meaning even if o and f are complex. It is easy to verify that D is proportional to c2 for the general case of o and f complex by differentiating (23) and using (10) and (22). Hence, the results are valid and for quantum systems. If the adiabatic invariant (23) contains a secular term, then the term can be eliminated by appropriate choice of the function S. In the paper the adiabatic invariants are given by the KBM variables. The transition to Hamiltonian variables q and p is given by (9). The invariants in q, p variables are of a strange structure. For example, the adiabatic invariant (41), for the Van der Pol oscillator and for o = 1 and b(t) = 0, is D,
=$(p’ + q2) + f
(p2+ q2MP2 + q2-4) arctang
-4P(P2
-q2+4) .
WV
Finally, let us consider several practical applications of the adiabatic invariants, a fact which considerably increases the importance of the theory. (a) Approximate solutions If we have two or more independent adiabatic invariants, for some problem, then we can _ construct an approximate solution of the corresponding differential equation of motion NLM 165,s- G
496
Dj.
S. DJUKK
(1). For example assuming the following initial conditions that CD=oO, tl=aO and 6 = 0 for t = 0, combining the adiabatic invariants (30) and (32) and using (13) we have an approximate solution of the problem in Section 4.1 a -=
a0
J(>( 00
l-
ti
.
&-o~
msln
c9
-l/Z
(45)
ti >
.
For small E. the solution is very close to another approximate solution, which can be obtained by the generalized method of averaging (see [ 11, p. 168). (b) Stability analyses The procedure for finding adiabatic invariants can be used for constructing Lyapunov’s function, that is, for stability analyses of the motion described by (1). Let us consider problem from the Section 4.1. Stability of the zero solution (4=4=0) of the corresponding differential equation (l), where f =O, is usually studied for 020 and ti G 0 (see for example [9]). According to the author’s knowledge, for 02 0 and W>, 0 there is no available result. Let us consider the following adiabatic invariant for the problem 2
sin 26 + bl m& , b, = constant >
(46)
which is obtained in a same way as the adiabatic invariant (32) but for So= b,aw’/(4m2). Here, b, is an arbitrary constant. The time derivative of 0: in virtue of the equations of motion (10) is ..
dT=(bl
2
-sin 2&z
ti2a2
- m
(2b, -2sin
26-b,
cos 26).
Selecting b, > 2/J3 we can see that the conditions of the Lyapunov’s stability theorem (see [29], pp. 222-225) are satisfied, i.e. the zero solution (q = 4 = 0) of the equation ( 1) is stable in Lyapunov’s sense, if w > 0, ti > 0 and
09
h
-
0
for t>,O
(48)
>O for b, >2/,/3.
(49)
where b*=2bl -\/(4+b:) 1 l+b,
If 61~0 for t 30 then the condition (48) is always satisfied. For cii>,O, the (ij must be in the interval 0 G &I< b:ti2/o for every r 2 0. The interval is widest for a large value of the constant b 1’
Performing a similar analysis, and starting with D,, equation (36), as the Lyapunov’s function we can prove that the motion of a pendulum of variable length around the vertical equilibrium position, is stable in Lyapunov’s sense if o>O, tia0.
c;j< -/3*ti2/o
for ta0
(50)
where p,=j(982+4)-2P,o P-l
(51)
for every fi selected such that /?> 1. For this problem the stability results exists for ti
(c) Phase curues The adiabatic invariants, for example equation (44), can be used for drawing the corresponding phase curves in the p. q plane or the a, 6 coordinate system. A graphical display device technique must be used in this case.
Adiabatm
mvartants
for dynamtcal
systems wrth one degree of freedom
497
Acknon~/e&emcn/~ The author IS grateful to Professor A. M Strauss for his constructive review of the paper. Also. dtscussrons with Professor M R M Crespo da Stlva and Professor C M. Hulley on the manuscript are very much apprectated.
I Ah Hasan Nayfeh. Perrurbu/ion Method\. John Wiley. New York (1973). 2. Dt S. Diukic. A procedure for finding first integrals of mechamcal systems with -gauge-vartant lagrangians, Ini. .I. N;rn-Line& Med. 8.479 (1973) 3 Dj S. Djuklc and B G. Vujanovic. Noether’s theory in classical nonconservative mechanics. Acfu Mechanica 4 5 6 7 8. 9 IO. II. 12. 13. 14. 15. 16. I7 I8 I9 20 21 22 23
23. 17 (1975) Dj. S DJukic. A new first integral corresponding to Lyapunov‘s function for a pendulum of variable length. 2. Anyew Murh Php 25.532 (I 974) Dj S. DJuktc. Conservatton laws in classical mechamcs for quasi-coordinates. Arch. Rotrod Mech. Andy. 56.79 (1974) B. D VuJanovic. A group-variattonal procedure for finding first integrals of dynamical systems. Inf. J NonLmeur Mech 5,269 (19701 B. D Vujanovtc. A geometrtcal approach to the conservation laws of non-conservative dynamical systems. 7mtor. 32, 357 (1978). B D. Vujanovic. Conservation laws of dynamical systems via D’Alembert’s princtple. fnr. J. Non-Lineur Med. 13, 185 (1978) G Leitmann. On the stability of solution of a non-linear non-autonomous equations. Inr J. Non-Lmeur Mech 1, 291 (1966) H R. Lewis. Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians, Phys Rec. Lerr. 18, 51Ot1967). M. Born. The Mechanics oj the Atom. p. 56. Frederick Ungar (1960). H. R. Lewts. Class ofexact invariants for classical and quantum time-dependent harmonic oscillators. J. Marh. Phyr. 9. 1976 (1968). H R. Lewis and W. B. Riesenfeld. An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a ttme-dependent electromagnetic field, J. Marh. Phys 10, 1458 (1969). D P. Stern. Kruskal’s perturbation method. J. Muth. Phys. 11,277l (1970). D. P. Stern. Classical adiabatic perturbation theory. J. Murh. Phy. 12, 2231 (1971). K R. Symon. The adtabattc invariant of the linear or nonlinear oscillator. J. Math. Phys. 11, 1320 (1970). J. Guyard. A Nadeau, G Baumann and M. R. Feix, Eigenvalues of the Hill equatton to any order in the adiabatic limtt. J. MU/I. Phys. 12, 488 (1971). G. E. 0 Giacaglia. Perfurburion Methods in Non-Linear Systems, p. 263. Springer (1972). D C. Khandekar and S. V Lawande, Exact propagator for a time-dependent harmonic oscillator with and without a singular perturbation. J. Muth. Phys. 16,384 (1975). W. Sarlet. Class of Hamiltons with one degree-of-freedom allowing application of Kruskal’s asymptotic theory in closed form I and II. Ann. Phys. 92, 232 (1975); ibid. 92, 248 (1975). N. J Gunther and P G. Leach. Generalized invariants for the time-dependent harmonic oscillator. J. Math. Ph)s. 18. 572 119771 P. G. Leach. On the theory of time-dependent linear canonical transformations as applied to Hamiltonians of the harmonic oscillator type. J. Moth. Phys. 18, 1608 (1977). W. Sarlet. Exact mvariants for time-dependent Hamiltonian systems w)or one degree-of-freedom, J. Phys. A. 11,843 (1978).
24
W. Sarlet. tnvartance and conservation /‘hp. 19, 1049 11978).
25 26 27
M Lutzky. Noethers theorem and the time-dependent harmonic oscillator. Phys. Letr. 68A, 3 (1978). J. M. Burgers. Die adiabatischen Invarianten bedingt periodischer Systeme. Ann. Physik 52, 195 (1917). M. Kruska). Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic. J. Moth. Phys. 3, 806 ( 1962).
28 29
Dj. S. DJukic, A contribution to the generalized Noether’s theorem, Arch Mech. 26.243 (1974). D. R Merkin. Inrroducrton to Stabiliry of Motion Theory tin Russian), Nauka. Moscow (1971).
Risumi
laws for Lagrangian
systems with one degree of freedom. J. Math.
:
On e’tabl
it
des
invariants
adiabatiques
pour
des
syst&nes
un degr6 de liberti dont I’iquation de mouvement est (1). et O?I on n’impose pas l’existence de 1’ intigrale d’act ion de Hami 1 ton correspondante. Les invariants adiabatiques peuvent varier selon leur strucEn utilisant cette th6orie on analyse quelques ture. problimes particuliers comprenant les oscillateurs non autonomes de Duffing et Van der Pol. Enfin, on indique comment on peut utiliser les invariants adiabatiques pour trouver des solutions approchies. pour des analyses de stabiliti et pour tracer des courbes de phase. dynamiques
2
Dj. S. DJIJKIC
498
Zusapnenfassung
:
Adiabatische Invarianten fiir dynamische Systeme mit einem Freiheitsgrad und einer Bewegungsgleichung nach (1). wobei das Vorhandensein des zugehiirigen Hamiltonischen integrals Die adianicht vorgeschreieben ist, werden aufgestellt. batischen Invarianten kijnnen nach ihrem Aufbau verschieden sein. Einige besondere Aufgaben-einschliesslich der nichtautonomen Schwinger nach Duffing und Van der Pol werden analysiert. Abschliessend wird angezeigt, wie die adiabatischen Invarianten zur Auffindung von N2herungslSsungen, in Stabilit6tuntersuchungen und zur Aufzeichnung von Phasenkurven benutzt werden kiinnen.
ADIABATIC INVARIANTS FOR DYNAMICAL SYSTEMS WITH ONE DEGREE OF FREEDOM DJ. S. DJUKIC~ Department of Engineering Science University of Cincinnati, Cincinnati, OH 45221. USA (Received 29 May 1979)
Abstract-Adiabatic invariants for dynamical systems with one degree of freedom. whose equation of motion is (I}, and where the existence of the corresponding Hamilton action integral is not imposed, are estabhshed. The adiabatic invariants may vary according to their structure. Using the theory a few particular problems, including non-autonomous Duffing and Van der Pol oscillators. are andysed. Finally, it is indicated how the adiabatic invariants can be used for finding approximate solutions, stability analyses and for plotting phase curves. I. INTRODUCTION
1n this paper, I propose to extend the idea of adiabatic invariants to dynamical systems with one degree of freedom, whose equation Of motion is &td(7)q=&f(z,
q, 4)
7=&f
(1)
indicates d/dt; ~, d2/dt2; ’ , d/&; and (I is the generalized coordinate; t is time, w is an arbitrary function of slowly varying time r, Eis a small parameter andfis an arbitrary function of the variables r, q and 4. By ‘adiabatic invariant’ for the dynamical system, we mean a function LI(C,7, q, 4) whose time derivative, by virtue of the equation of motion (1 X contains only terms higher than some particular order of the parameter E. If the time derivative of the function D is equal to zero then the function is an exact invariant, i.e. first integral, for the equation of motion (1). The basic results on adiabatic invariants are perhaps best stated in the work of Burgers [26]. In the last ten years, after Kruskal’s paper [27], there has been a growing interest in finding adiabatic (see [ 12-163) and exact (see [lo], [12], [13], [16], [20-231) invariants. All of the results are valid if the equation (1) can be derived from the corresponding Hamilton’s variational principle. If the dynamical system is purely non-conservative (for example the Van der Pol oscillator), that is, when the system cannot be described by H~iltonian canonical equations, the existing theory fails. Usually, adiabatic invariants are obtained by approximately satisfying the appropriate necessary conditions for the existence of the exact invariants. In fact, the adiabatic invariants form a series in powers of the parameter E. The leading term in the series is Born’s adiabatic invariant [l 11, i.e. it is energy divided by frequency. The central feature of the present discussion is the establishment of abiabatic invariants for the dynamical system (l), but where it is not assumed that the corresponding Hamiltonian canonical formalism exists. The theory is based on Noether’s theory for classical non-conservative mechanics [3] (for an extended review of the Noether’s theory consult [2-g]) and the use of Krylov-Bogoliubov-Mitropolski (KBM) asymptotic technique (see for example [l]). The adiabatic invariants obtained are series with respect to the small parameter a. Although the procedure outlined here is straightforward in principle, it is difficult to carry the procedure to a higher order without a prohibitive amount of labor. For this reason, the calculation of adiabatic invariants is concluded with the first order terms with respect to E. At the same time, it means that the time derivative Of the adiabatic invariants is proportional to &2,Here, the leading term in the adiabatic invariant series may be of various structures. Therefore, even if the dynamical system is a where
*On leave from the University of Novi Sad, 21000 Novi Sad, Yugostavia. 489
490
Dj S DJUKIC
Hamiltonian canonical system, the paper gives a new insight into the nature of the adiabatic invariant series. The general results of the paper can be easily applied to particular problems. We therefore restrict our attention to problems which are new or which illustrate particular features of the present adiabatic invariants. Typically, adiabatic invariants are used for practical purposes in quantum mechanics (see for example [ 173 and [20]). In this paper, it is indicated how adiabatic invariants may be used in vibration theory, stability analyses and for the construction of phase curves. 2
NOETHERlAN
THEORY
IN
KRYLOV-BOGOLIUBOV-WITROPOLSKI VARIABLES
To attack the problem of finding the adiabatic invariants, for the dynamical system (l), we must slightly generalize the results of the [3) and [28]. It is obvious that we may construct the Lagrangian and Hamiltonian function, for the conservative part [left-hand side of the equation (l)] of the system, in the following form
L=f[qZ -d(.r)q2];
H=gJ?+d(r)q2];
pL$j
(2)
where p is generalized momentum. Let us consider a continuous one-parameter transformation of the time, generalized coordinate, generalized velocity and generalized momentum ? 4, P)
f=:t+P&
4 “4 + I446 L 4, PI
p$f+r[i-qS+#t,
79%PJI
(3)
where p is a small parameter of the transformation and JI, 4, @ and v are functions of time, slow time, generalized coordinate and generalized momentum. Thus, corresponding to (3) there exists an infinitesimal transformation of the form At =p$, Aq =p$, A4 z/.+$-4$+0)
Ap Z/N.
(4)
Further, let us assume that the following expression V=(pd-H)
dt
(5)
is invariant in the sense that
(6)
1
AV= p$! -B(r, 4. p) df-[pi--H(t, q, p)] dt=O C when it is the object of the transformation. Combining (3), (4) and (6), developing the term B in series around the point t, q, p and retaining only members linear in the small parameter p, we arrive at the following expression
(7) which after a simple manipulation AV=p
becomes
-&#+H)+(cj-
-F/($-q$)+($+g
Now, let us consider the transformation KBM variables cxand 6, defined by q=a(t)
(8) of the Hamiltonian
cos h(t), p= -aw
sin 6.
variables q and p into the (9)
Adiahatx
InvarIants
tor dynamical
systems with one degree of freedom
491
Thus the original second-order differential equation ( 1), where 4 = p, for q may be replaced (see more details in [I]. p. 165) by the following two first-order differential equations 2 = - 2 EUJ’sin’ d - 5 f(5, r Cos 6. -20) sin d;)sin 6 flJ 0) 8=OJ - .fi 01’ sin b cos b - 2- f( 310 0
7, 2 cos 6
-
(10)
ZOJ sin 6) cos 6
for the KBM variables. where L’)‘= d&dr. At the same time, using (2) and (9) the equations (7) and (8) reduce to AV = ~1c v(r;icos 6 - ub sin b + m sin 6) -cm sin S(d + 0) L
- f$zw2 cos b - *EoJ~~~ -ZOJC$
1
a2uJ2
cos’ 6 - Ic; -
2
dt
(11)
r2u12 + icosd-aasin6 sin S-tf5 2 >(
+ 20~ sin d [Y+ (io sin b + ~oJ’&sin 6 + sod
COS S)$]
>
+(-iwsind-m)‘Esinb-rw6’cosS+aw2cos6-i$) x [(ti cos 6 - zd sin 6)$ - 4]-
(ao sin 6)@
--~f[&-_(icosd-rdsin6)$]
dt
(12)
I while the Hamiltonian
(2) becomes r
H=F.
W
(13)
Remembering that the dynamical system under consideration moves in agreement with the equations (10). and assuming that the functions @.4 and tj satisfy the algebraic equation rwdr sin d = --~f($ + ro$ sin b)
(14)
we may deduce the following theorem from (6) and (12) Theorem 1. If. under the continuous infinitesimal one-parameter transformation (4). which satisfies the algebraic equation (9). the expression (5) is invariant in the sense of equation (6). then the quantity D=rwr$
r20J2 sin 6+--2 *
(15)
is constant of motion. i.e. it is an exact invariant for the dynamical system under consideration. Now. combining (6). (10). (11) and (14) we obtain the necessary condition for the functions 4 and Ic/.that must be satisfied for the function V to be invariant under the infinitesimal transformation (4) @to’ Cos b + fjrw sin 6 + *wo’Ea2 cos2 6 x21d +-- 2 Icj-_Es(zw$ sin 6+$)=0
(16)
while the function Q, must satisfy the algebraic equation (14). Here we may remark that after integration of the equation (16) with respect to 4 and $ the function @ can be easily found from equation (14). Also, from this analysis, one can establish the next theorem. Theorem 2. If the equation ( 16) admits a solution in 4 and $, then the equation of motion (1) admits an exact invariant ( 15) in the KBM variables. The theorem is in fact a recipe for finding the exact invariants. It is important that the equations (15) and (16) define a class of
49’
DJ S DJUKK
exact invariants because 4 and $ may be any particular solution of equation (16). Here, we make one important remark. The solution of the equation (16) and the first integral (15) do not depend on the quantity v, which appears in the transformation law for the generalized momentum (3). At first it may appear that this quantity may take arbitrary values. It can be seen that this supposition is erroneous if we recall that in Hamiltonian mechanics the generalized momenta are known functions of time, generalized coordinates and generalized velocities. Hence, the transformed generalized momentum p is completely determined by the transformed quantities r, 4 and dg/dr, i.e. the quantity v can be obtained as a function of $, 4 and Q, [see (3)]. Thus the logic of Hamilton’s mechanics is maintained. 3
THE
ADIABATIC
INVARIANTS
The class of exact solutions of the equation (16), i.e. the corresponding class of exact invariants, is extremely limited by a particular choice of the functions o andfin differential equation (1). For this reason, we will concentrate our attention on the approximate solutions of the equation (16). At the same time, it means that the invariance condition for the expression V, that is, that AV = 0, will not be satisfied. Namely, we will search for such solution of the equation (16) for which the AV is proportional to s2. It will lead us to the adiabatic invariants for the dynamical system (1). Let us consider the following series $ = Jl&, w, 6, +=&(a,04
7, t) + EJ/ I(a, w, 6, T, t, 0’) + c2.
6, r, t)+i$r(a,
048, T, t,o’)+e2.
..
(17)
..,
(18)
with respect to the small parameter E, as a solution of the equation (16). Here, the eO, $1,. . * 7 $0, 41,*.. 9 are unknown functions of the variables indicated in the brackets. Substituting (17) and (18) into the equation (16), using (10) to eliminate oi and 8 from the obtained result and equating with zero the terms in corresponding powers of E, because residual AV of the equation (16) has to be proportional to e2 for every E,we have the following two equations &o cos 6+ rk-++
?$)sin~+~(C!$!?+~~)=O
34, 84, sin 6 o~+c~‘~+~+~
4,ao2 cosS+ao
(19)
a+, 84,
i
_;C!&
sin6+Tw’(l-cos26)]-Aa%
-!fcos~? ( a2m2
+kw’.sin26
%o 0’;;;;;
+~$pd(l+COS26)+~ >I
i
_ wo
1i(ljo 1 --( +w$+g+-
w&5
?r
-
kg
-fcosS+Gsin26 u
= c$J+ aw$,f
_I
fsin 6+:0’(1
1
-cos 26)
[ >I
sin 6.
(20)
Solving these equations with respect to &,,4r, $0 and $1
wehave (21)
where
T, a cos 6, -am + Sd7,
a, w, w’)
sin 6) sin 6d 6
(22)
Adiabatic invariants for dynamxal systems with one degree of freedom
493
and where S and S, are arbitrary functions of the corresponding variables. Here, we must underline the fact that the solution (21) is not a unique solution to the equations (19) and (20). Now, ~mbining (17), (18) and (21) with (1 St we have the quantity D as I) = aoS(a, w, t) + EawF(a, u, T, 6, m’)
(23)
whose time derivative by virtue of the equations of motion (10) is
- (F+ag)[f
sin S+iw’(l
-cos 26)] (24)
Therefore, the quantity D may be called an adiabatic invariant for the dynamical system whose equation of motion is given by (1) or (10). Also, using (4), (14), (17), (18) and (21) we have the infinitesimal transformation in the form
under which the function V, given by (5), has the change of the order Em.For this reason, we may say that the V is adiabatic invariant when it is the object of the transformation of variables (25) and (26). Therefore, the existence of the adiabatic invariant D is caused by the adiabatic invariance of the function V. Using (13), the leading term awS in the adiabatic invariant (23) becomes
where the function S is still arbitrary. Hence, the term may be of a general structure, which may be completely different than the energy (H) divided by the frequency @I) as it is the case in the existing literature. This gives a new insight into the nature of adiabatic invariants. Now, let us suppose that the functions S and S, are of the following form S=ka*o” 7S o--0
CW
where k, b and c are arbitrary constants. Selecting particular values for these constants and using (13), (22) and (23) we have some special types of the adiabatic invariants.
(a) Classical type For k = l/2, b = 1, c = 0 we have an adiabatic invariant DI=E
--E sfksin26
(28)
where the leading term is energy divided by frequency.
(b) Energytype Fork=1/2,b=l
andc=l D,=H-+
wehave &
H(26+sin
,/(2H) 2s) - 0
i.e. the adiabatic invariant with energy as the leading term.
f sin 6dS f
1
Dj. S. DJUKIC
494
(cl 0, S type For k = 1, b = - 1 and c = 1 we have the following adiabatic invariant D, = 0’ - 2dEb,
(30)
which is a function of the w and 6 only.
4
APPLICATIONS
The above general results can easily be applied to a particular problem. The application requires calculation of the integral f f sin 6 da, for a particular structure of J Obviously, the functionfmust be expressed in the KBM variables. It is interesting that the adiabatic invariant D,, which is given by (30), is of the same structure for every& For the reason that the adiabatic invariants are approximately constants of motion, the ‘numbers’ D1, D2, . . . can be calculated by the initial conditions. 4.1. Classical time-dependent harmonic oscillator As the first example, let us consider the classical time-dependent harmonic oscillator. In this case the differential equation of motion is given by (l), where j-=0. The corresponding
(31)
adiabatic invariants of classical (28) and energy (29) type are D 1=- H w
(
sin 26 , (ti=m’)
1-s
>
D,=H
l-
6
1.
(26 + sin 26)
[
(3.2)
(33)
While the second adiabatic invariant D2 is unknown in the literature, the first one differs from those proposed by Symon [16] for the following term H o’2&2 cos2 6. 405
The term and its time derivative are of the order .s2. Hence, the qualities of the Symon’s invariant and the D, are the same. 4.2. Pendulum of variable length The equation of motion of a simple pendulum with small angular displacement is of the form (1) with
Now, the classical type adiabatic invariant (28) is D, =z+Eg(3
sin 26-86).
In the next section of the paper, it will be indicated how the adiabatic invariants can be used in stability analyses. For this purpose, the following invariant for the same problem is much more convenient D,=a20-3+~w-Su@+sin
26).
The result is obtained by making use of (13), (22) and (23) and for S=aoe4,
where j? is constant.
3a
SO=B~~’
(36)
Adiabatic
mvariants
for dynamlcal
systems with one degree of freedom
495
4.3. Non-autonomous Dufflng equation
In this case the characteristic
functionfis f=bo(r)q3+W)
(38)
where b, and bl are arbitrary functions of the slowly varying time T. The adiabatic invariant of the classical type (28) is H W’H sin 26 - Ao2 D, =w--E~
y
H2 cos4 6 +J(2H)b,(r)
cos 6
1
(39)
4.4. Non-autonomous Van der Pol oscillator For the governing differential equation of motion 4 + w2(z)q = E[( 1 -q2)4 + b(r)] where b is an arbitrary function of
7,
(40)
the classical type adiabatic invariant is
+ 4H sin 26 - 8,/(2H)b(r)
cos 6 -
.
(41)
4.5. An oscillator with quadratic non-linear term If the characteristic function of the problem is
f =b(r)q2
(42)
where the b is an arbitrary function of r, the equation (28) yields the following adiabatic invariant --E&Hsin26+
2 2 3. b(r)H3’2 cos3 6 -JX
(43)
whose time derivative is of s2 order. For the same problem Symon [16] has obtained an adiabatic invariant, but whose time derivative is of the order E. 5.
DISCUSSION
In this paper I have proposed a procedure for finding adiabatic invariants for a dynamical system of one degree of freedom, whose equation of motion is given by (1). The results obtained are very general and valid for any structure of the ‘non-conservative force’J Although it is assumed that the system is a classical dynamical system, that is, that o and f are real, the adiabatic invariant D has the same meaning even if o and f are complex. It is easy to verify that D is proportional to c2 for the general case of o and f complex by differentiating (23) and using (10) and (22). Hence, the results are valid and for quantum systems. If the adiabatic invariant (23) contains a secular term, then the term can be eliminated by appropriate choice of the function S. In the paper the adiabatic invariants are given by the KBM variables. The transition to Hamiltonian variables q and p is given by (9). The invariants in q, p variables are of a strange structure. For example, the adiabatic invariant (41), for the Van der Pol oscillator and for o = 1 and b(t) = 0, is D,
=$(p’ + q2) + f
(p2+ q2MP2 + q2-4) arctang
-4P(P2
-q2+4) .
WV
Finally, let us consider several practical applications of the adiabatic invariants, a fact which considerably increases the importance of the theory. (a) Approximate solutions If we have two or more independent adiabatic invariants, for some problem, then we can _ construct an approximate solution of the corresponding differential equation of motion NLM 165,s- G
496
Dj.
S. DJUKK
(1). For example assuming the following initial conditions that CD=oO, tl=aO and 6 = 0 for t = 0, combining the adiabatic invariants (30) and (32) and using (13) we have an approximate solution of the problem in Section 4.1 a -=
a0
J(>( 00
l-
ti
.
&-o~
msln
c9
-l/Z
(45)
ti >
.
For small E. the solution is very close to another approximate solution, which can be obtained by the generalized method of averaging (see [ 11, p. 168). (b) Stability analyses The procedure for finding adiabatic invariants can be used for constructing Lyapunov’s function, that is, for stability analyses of the motion described by (1). Let us consider problem from the Section 4.1. Stability of the zero solution (4=4=0) of the corresponding differential equation (l), where f =O, is usually studied for 020 and ti G 0 (see for example [9]). According to the author’s knowledge, for 02 0 and W>, 0 there is no available result. Let us consider the following adiabatic invariant for the problem 2
sin 26 + bl m& , b, = constant >
(46)
which is obtained in a same way as the adiabatic invariant (32) but for So= b,aw’/(4m2). Here, b, is an arbitrary constant. The time derivative of 0: in virtue of the equations of motion (10) is ..
dT=(bl
2
-sin 2&z
ti2a2
- m
(2b, -2sin
26-b,
cos 26).
Selecting b, > 2/J3 we can see that the conditions of the Lyapunov’s stability theorem (see [29], pp. 222-225) are satisfied, i.e. the zero solution (q = 4 = 0) of the equation ( 1) is stable in Lyapunov’s sense, if w > 0, ti > 0 and
09
h
-
0
for t>,O
(48)
>O for b, >2/,/3.
(49)
where b*=2bl -\/(4+b:) 1 l+b,
If 61~0 for t 30 then the condition (48) is always satisfied. For cii>,O, the (ij must be in the interval 0 G &I< b:ti2/o for every r 2 0. The interval is widest for a large value of the constant b 1’
Performing a similar analysis, and starting with D,, equation (36), as the Lyapunov’s function we can prove that the motion of a pendulum of variable length around the vertical equilibrium position, is stable in Lyapunov’s sense if o>O, tia0.
c;j< -/3*ti2/o
for ta0
(50)
where p,=j(982+4)-2P,o P-l
(51)
for every fi selected such that /?> 1. For this problem the stability results exists for ti
(c) Phase curues The adiabatic invariants, for example equation (44), can be used for drawing the corresponding phase curves in the p. q plane or the a, 6 coordinate system. A graphical display device technique must be used in this case.
Adiabatm
mvartants
for dynamtcal
systems wrth one degree of freedom
497
Acknon~/e&emcn/~ The author IS grateful to Professor A. M Strauss for his constructive review of the paper. Also. dtscussrons with Professor M R M Crespo da Stlva and Professor C M. Hulley on the manuscript are very much apprectated.
I Ah Hasan Nayfeh. Perrurbu/ion Method\. John Wiley. New York (1973). 2. Dt S. Diukic. A procedure for finding first integrals of mechamcal systems with -gauge-vartant lagrangians, Ini. .I. N;rn-Line& Med. 8.479 (1973) 3 Dj S. Djuklc and B G. Vujanovic. Noether’s theory in classical nonconservative mechanics. Acfu Mechanica 4 5 6 7 8. 9 IO. II. 12. 13. 14. 15. 16. I7 I8 I9 20 21 22 23
23. 17 (1975) Dj. S DJukic. A new first integral corresponding to Lyapunov‘s function for a pendulum of variable length. 2. Anyew Murh Php 25.532 (I 974) Dj S. DJuktc. Conservatton laws in classical mechamcs for quasi-coordinates. Arch. Rotrod Mech. Andy. 56.79 (1974) B. D VuJanovic. A group-variattonal procedure for finding first integrals of dynamical systems. Inf. J NonLmeur Mech 5,269 (19701 B. D Vujanovtc. A geometrtcal approach to the conservation laws of non-conservative dynamical systems. 7mtor. 32, 357 (1978). B D. Vujanovic. Conservation laws of dynamical systems via D’Alembert’s princtple. fnr. J. Non-Lineur Med. 13, 185 (1978) G Leitmann. On the stability of solution of a non-linear non-autonomous equations. Inr J. Non-Lmeur Mech 1, 291 (1966) H R. Lewis. Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians, Phys Rec. Lerr. 18, 51Ot1967). M. Born. The Mechanics oj the Atom. p. 56. Frederick Ungar (1960). H. R. Lewts. Class ofexact invariants for classical and quantum time-dependent harmonic oscillators. J. Marh. Phyr. 9. 1976 (1968). H R. Lewis and W. B. Riesenfeld. An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a ttme-dependent electromagnetic field, J. Marh. Phys 10, 1458 (1969). D P. Stern. Kruskal’s perturbation method. J. Muth. Phys. 11,277l (1970). D. P. Stern. Classical adiabatic perturbation theory. J. Murh. Phy. 12, 2231 (1971). K R. Symon. The adtabattc invariant of the linear or nonlinear oscillator. J. Math. Phys. 11, 1320 (1970). J. Guyard. A Nadeau, G Baumann and M. R. Feix, Eigenvalues of the Hill equatton to any order in the adiabatic limtt. J. MU/I. Phys. 12, 488 (1971). G. E. 0 Giacaglia. Perfurburion Methods in Non-Linear Systems, p. 263. Springer (1972). D C. Khandekar and S. V Lawande, Exact propagator for a time-dependent harmonic oscillator with and without a singular perturbation. J. Muth. Phys. 16,384 (1975). W. Sarlet. Class of Hamiltons with one degree-of-freedom allowing application of Kruskal’s asymptotic theory in closed form I and II. Ann. Phys. 92, 232 (1975); ibid. 92, 248 (1975). N. J Gunther and P G. Leach. Generalized invariants for the time-dependent harmonic oscillator. J. Math. Ph)s. 18. 572 119771 P. G. Leach. On the theory of time-dependent linear canonical transformations as applied to Hamiltonians of the harmonic oscillator type. J. Moth. Phys. 18, 1608 (1977). W. Sarlet. Exact mvariants for time-dependent Hamiltonian systems w)or one degree-of-freedom, J. Phys. A. 11,843 (1978).
24
W. Sarlet. tnvartance and conservation /‘hp. 19, 1049 11978).
25 26 27
M Lutzky. Noethers theorem and the time-dependent harmonic oscillator. Phys. Letr. 68A, 3 (1978). J. M. Burgers. Die adiabatischen Invarianten bedingt periodischer Systeme. Ann. Physik 52, 195 (1917). M. Kruska). Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic. J. Moth. Phys. 3, 806 ( 1962).
28 29
Dj. S. DJukic, A contribution to the generalized Noether’s theorem, Arch Mech. 26.243 (1974). D. R Merkin. Inrroducrton to Stabiliry of Motion Theory tin Russian), Nauka. Moscow (1971).
Risumi
laws for Lagrangian
systems with one degree of freedom. J. Math.
:
On e’tabl
it
des
invariants
adiabatiques
pour
des
syst&nes
un degr6 de liberti dont I’iquation de mouvement est (1). et O?I on n’impose pas l’existence de 1’ intigrale d’act ion de Hami 1 ton correspondante. Les invariants adiabatiques peuvent varier selon leur strucEn utilisant cette th6orie on analyse quelques ture. problimes particuliers comprenant les oscillateurs non autonomes de Duffing et Van der Pol. Enfin, on indique comment on peut utiliser les invariants adiabatiques pour trouver des solutions approchies. pour des analyses de stabiliti et pour tracer des courbes de phase. dynamiques
2
Dj. S. DJIJKIC
498
Zusapnenfassung
:
Adiabatische Invarianten fiir dynamische Systeme mit einem Freiheitsgrad und einer Bewegungsgleichung nach (1). wobei das Vorhandensein des zugehiirigen Hamiltonischen integrals Die adianicht vorgeschreieben ist, werden aufgestellt. batischen Invarianten kijnnen nach ihrem Aufbau verschieden sein. Einige besondere Aufgaben-einschliesslich der nichtautonomen Schwinger nach Duffing und Van der Pol werden analysiert. Abschliessend wird angezeigt, wie die adiabatischen Invarianten zur Auffindung von N2herungslSsungen, in Stabilit6tuntersuchungen und zur Aufzeichnung von Phasenkurven benutzt werden kiinnen.