Resonant modification and destruction of adiabatic invariants

ANNALS

OF PHYSICS:

71, 319-356 (1972)

Resonant Modification

and Destruction

of Adiabatic

Invariants”

E. F. JAEGER AND A. J. LICHTENBERG Department

of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, California 94720

Received April 20, 1971

For a harmonic oscillator with time varying coefficients a relation is obtained between the action integral and the tirst order adiabatic invariant. It is shown that a small resonant perturbation can modify the invariant. A canonical transformation generates a new action variable constant to first order in the perturbation with the old actionangle variables playing the role of oscillatory momentum and extension. There are two separate ways in which the new adiabatic constant can break down, leading either to a modified invariant or to a situation in which no invariant exists. A general theory is developed to use the procedure for multidimensional systems. It is shown that two distinct types of resonances are possible that lead to qualitatively different results. The general theory is applied to a coupled oscillator system in forms demonstrating both types of resonances. A canonical transformation is used to remove the more important type of resonance, and an average is performed over a time comparable to the slower period of the oscillator. Numerical integrations of both the averaged and exact Hamiltonian equations show the maximum value of the perturbation for which the averaging process is valid. The system is found to oscillate slowly about resonance but, as long as the averaging process is valid, this slow oscillation is adiabatically separated from the faster motion of the oscillator by the existence of an adiabatic invariant. When averaging is not valid, a higher order resonant coupling occurs between the slow drift about resonance and the motion of the oscillator. The canonical transformation as described in the general theory is made near a particular higher order resonance to obtain the oscillation in the adiabatic invariant. For oscillations about neighboring resonances which do not interact strongly a new approximate invariant exists. For strong interaction the breakup of the invariant curves is demonstrated and the strength of the perturbation necessary for breakup is compared with a simple analytic criterion.

* The work is partly based on a Ph.D. thesis of E. F. Jaeger. A. J. Lichtenberg was a Miller Professor and E. F. Jaeger an AEC Fellow and an NSF Fellow during part of this research. The work was also partially supported by the National Science Foundation under Grant GK2978, and the Air Force Office of Scientific Research under Grant AF-AFOSR-69-1754, and by the University of California at Berkeley Computer Center. 319 Copyright AI1 rights

0 1972 by Academic Press, Inc. of reproduction in any form reserved

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I. INTRODUCTION For a one dimensional oscillator in which the Hamiltonian is a slowly varying function of time, the action integral, J = $p dq is an approximate constant or “adiabatic invariant” of the motion. Here p and q are canonical coordinates for the momentum and position and the integration is carried over a complete period of the oscillation. If the Hamiltonian is initially constant, then slowly varying with time, and finally constant again, the action integral is constant to all orders in an expansion parameter, E [l-3]. These results do not mean that J is an exact invariant but only that the total time derivative of J approaches zero faster than any power of E. An iterative technique has been used to calculate the change in the magnetic moment of a particle in a magnetic field 141.Vandervoort [5] uses a similar iterative method to calculate the change in the action integral for the harmonic oscillator. An exponentially small change in the action is obtained which is not inconsistent with the asymptotic result. Changes in the action can also be calculated using the asymptotic method if discontinuities are assumed in higher order derivatives of the time varying parameter [6]. These discontinuities approximate changes in the derivatives which result from the finite rate at which the Hamiltonian varies. The unperturbed action is the first term of the asymptotic series which gives the total adiabatic invariant for the time varying problem. As we shall see in Section II, if the next term in the series is included, its time rate of change exactly cancels the change in the first term as calculated by Vandervoort. This cancellation occurs to each successive order in the series, and it is not until a term including a “discontinuous” derivative is reached that the asymptotic series breaks down. If the slow time dependence of the Hamiltonian is periodic, a rapidly varying derivative is equivalent to a resonance between a harmonic of the slow time dependence and the oscillator frequency, which destroys the adiabatic invariance of the action integral in that order. In practical cases only the lowest order resonances are strong enough to have significant effect. For harmonic oscillator exact invariants also exist [7]. For nonlinear oscillators a similar invariant can be obtained by expansion in the nonlinearity [B], but in the case of periodic coefficients the expansion may not converge. By properly choosing the constants of integration the exact invariant can be chosen equal to the adiabatic invariant at any initial time. Then, by comparing the two invariants at a later time, an estimate is made of the change in the adiabatic invariant. If the adiabatic invariant is destroyed due to resonances then we expect the “exact” invariant to be destroyed also. For multidimensional problems such as nonlinearly coupled oscillators, the motion of charged particles in electric and magnetic fields, and the restricted three body problem, if the oscillation in one degree of freedom is very much faster than the motion in the remaining degrees of freedom, then the rapid oscillation can be

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isolated from the remaining motion by treating it as a one dimensional oscillator with slowly varying parameters. Bogoliubov and coworkers [g-10] developed a “method of averaging”, for the rapidly rotating phase associated with the fast oscillation, which gives the motion of the slow drift alone, dependent only on the adiabatic invariants of the fast motion. Kruskal [ll] developed a method similar to that of Bogoliubov demonstrating that the adiabatic invariant is constant to all orders in the expansion parameter. McNamara and Whiteman [12, 131 use a simpler procedure to calculate higher order adiabatic invariants for multidimensional problems and they show their results are equivalent to those of Kruskal. Contopoulos [14] has pointed out that, for Hamiltonians which are periodic in time, two distinct forms of invariants can be constructed. The second form is known as a third integral because of its use in dynamical problems having two known integrals. Although both adiabatic invariants and third integrals are expansions in a small parameter, the small parameter in the third integral case refers to the small term in the Hamiltonian while in the adiabatic invariant case, it refers to a slow dependence of the Hamiltonian on a particular variable. For a one dimensional problem in which both expansions can be calculated, Contopoulos has shown the adiabatic invariant to be better conserved than the third integral when the Hamiltonian varies slowly with time. When the time variation is not slow but the perturbation term is small the third integral is better conserved. He also notes that if a resonance exists between the perturbation and the frequency of oscillation, both expansions fail due to secularities. It is the effect of resonances between harmonics of the slow oscillations and the fast oscillation in modifying and destroying adiabatic invariants that is the main subject of this paper. In multidimensional systems, an invariant may be an “isolating integral” in that it separates two degrees of freedom. It is well known that, in multidimensional problems, there are transitions between regions in which isolating integrals exist and regions in which they do not [15-171. The technique for observing these transitions is to look at the crossings of the trajectory in a surface of section corresponding to a particular phase of the most rapid oscillation. If the crossings lie on a smooth curve, the motion in that plane is isolated from the rest of the motion, and the isolating integral exists. If the crossings are random the integral does not exist. The computations indicate that transitions between smooth and ergodic trajectories often are accompanied by the breaking of a single curve into a number of discrete curves or islands. From the island structure we can identify the order of the resonance. The numerical results and their relationship to the destruction of invariants have been summarized by Lichtenberg [17]. Chirikov [18] considers resonances between the Larmar rotation and the longitudinal bounce of a particle in a magnetic mirror and calculates the rate of change of the magnetic moment p due to the resonance. Because of nonlinearities the system does not remain in resonance, and p oscillates about its resonant value. A criterion

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for destruction of the invariant proposed by Chirikov is that the values of the action from the two adjacent resonances overlap. Similar overlap criteria have been used for the problem of the destruction of magnetic surfaces [19,20], and for a study of the stability of the Korteweg-deVries equation [21]. None of the above results of resonant destruction of invariants has been verified by numerical computations of the exact equations of motion. However, exact orbit calculations for electron-cyclotron heating in a magnetic mirror have demonstrated that the destruction of the invariants occurs near a resonance between two degrees of freedom of the system [22]. Walker and Ford [23] have also predicted the onset of invariant destruction for some simple two dimensional oscillator systems. They calculate the value of the perturbation at which the separatrices of neighboring resonant oscillations overlap and obtain good agreement with numerical solutions. Their oscillator systems only contain the resonant terms, however, and therefore the effect of other harmonics of the nonlinear oscillators is not revealed by their calculation. In addition to the fact that the resonance width can only be calculated approximately, the concept of overlap is itself approximate. It is not obvious how close together resonances must be for breakdown to occur. In fact, we shall show with numerical examples that the interaction of the resonant terms with the nonresonant terms can lead to destruction of the invariant even if the resonances are well separated. Nevertheless, the overlap criterion is correct within an order of magnitude, and for simplicity we shall use the term “resonance overlap” to indicate invariant destruction, with the implicit notion that “overlap” means a strong interaction between neighboring resonances. Another complication, is that the primary resonance in the system is not always the cause for the breakdown of the invariant. Instead, the primary resonance may generate islands that have harmonic frequencies creating secondary resonances, and these secondary resonances may cause the breakdown. In fact, there is a hierarchy of resonances generated in the multiply periodic system and invariant destruction can, in principle, occur at any level of the hierarchy. The removal of a degeneracy in two degrees of freedom, with a low order resonance, does not generally lead to overlapping resonances, except near the separatrix of the oscillations. The destruction of the invariant in these cases is usually caused by secondary resonances. This has led to some confusion in the work on the breakup of magnetic surfaces in toruses. For the adiabatic problem, on the other hand, the resonance destruction is most likely caused by “overlap” of the resonances caused by two adjacent harmonics of the slow oscillation. The destruction of adiabatic invariants due to resonances is related to the problem of small denominators in classical perturbation theory. This problem has occupied mathematicians since the time of Poincare [24], and there have been many methods devised to correct the perturbation technique so that terms with small denominators do not occur to destroy the convergence of the series. One such

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method which we use in subsequent sections is degenerate perturbation theory or the method of secular perturbations [25]. The resonant or degenerate variables are eliminated from the unperturbed Hamiltonian by a canonical transformation to a frame of reference that rotates with the resonant frequency. The new coordinates then measure the slow oscillation of the variables about their values at resonance, which is an elliptic fixed point of the new phase plane. This technique has been applied to the nonlinear theory of accelerator dynamics (see [26] for treatment and additional references) and to electron cyclotron resonance in a mirror field [27]. The procedure is also similar to that used by Chirikov [18] except that here canonical variables are preserved throughout the transformations. The adiabatic procedures can be used to average over the rapidly rotating phase after the resonance has been removed. However, if E is not small enough, higher order accidental degeneracies may occur to destroy the new adiabatic invariant. These higher order resonances can also be removed by transforming to new coordinates that rotate with the resonant frequency. Under certain conditions, the system may remain close enough to the higher order resonance that averaging can be performed a second time. In this case, modified adiabatic invariants are found which govern the oscillation in the previous adiabatic invariant about its value at resonance. The existence of resonances is closely linked to the convergence of the asymptotic series in multidimensional systems. For almost periodic solutions represented as motion on an n-dimensional torus and for nonlinear coupling that is sufficiently small with frequencies that are sufficiently incommensurable, the effect of the perturbation is only to slightly deform the toroidal surfaces. But for frequencies that are commensurable, the perturbed tori are deformed greatly and the particle orbits do not remain close to the unperturbed torus [28-301. One application of this theorem is the proof of the eternal invariance of the action for the one dimensional oscillator with slow periodic variation of the Hamiltonian. However quantitative measures of sufficient incomensurability and sufficient slowness are difficult to obtain from these theorems. The radius of convergence of the series is related to the presence of the interacting resonant terms, and can at present only be estimated from numerical studies. The procedure for determining the invariants of a multidimensional system is rather involved, and we here outline the general steps as they will be used in the following sections: (1) The Hamiltonian is divided into two parts, a zero order part which can be transformed to action angle variables and a first order part (in some small parameter, E) which cannot be transformed. (2) (a) If one frequency of the unperturbed system is much faster than the others, the method of averaging is employed to obtain the first order invariant of the fast oscillation, including the perturbation. (b) If two frequencies have a low order commensurability a canonical transformation is made to a rotating frame in which there is only a single frequency for the unperturbed motion (intrinsic degeneracy) or two widely spaced frequencies

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(accidental degeneracy). The averaging can then be performed as in (a). (3) If after the transformation a resonance persists between a fairly low harmonic of the slow variable and the fast variable, the averaging process is not valid, and the additional resonance must be removed, leading to an “island oscillation.” However, the transformed variables in the rotating frame are no longer in action-angle form. The Hamilton-Jacobi equation must be solved to reintroduce action variables. This is generally done by a four step process. (i) The Hamiltonian is averaged over the fast variable. (ii) The averaged Hamiltonian is expanded about an elliptic singular point and action angle variables are obtained to lowest order in the expansion. (iii) Perturbation theory is used to obtain the action-angle variables to higher order for the averaged Hamiltonian. (iv) The angle dependent terms are then reintroduced to obtain a Hamiltonian as in (1). (4) The process in (2b) is then repeated for the island oscillation to obtain the new invariant of the motion.

II. THE ONE DIMENSIONAL

OSCILLATOR

WITH TIME VARYING

FREQUENCY

We consider here a one-dimensional example to illustrate the physical effect of resonances on the adiabatic invariance of the action integral. A resonance can be observed in one dimension by allowing a slow periodic time dependence in the Hamiltonian. Commensurability between harmonics of the slowly varying parameter and the frequency of the one-dimensional oscillation generates resonances similar to those in multidimensional systems. The problem is somewhat simpler in that the slowly varying frequencies are constants, not dependent on the value of an action. To limit the amount of algebra in this section, we introduce certain canonical transformations intuitively rather than deriving them from generating functions. These transformations will be derived more rigorously in later sections but here we concentrate more on the physical meaning of the variables. Consider the Hamiltonian H = (p2/2m) + (mw2q2/2),

(2.1)

where p, q, and m denote momentum, position, and mass, respectively, for an oscillating particle. The frequency of oscillation w is assumed to be a slowly varying function of time: w = o(d). (2.2) We define new variables, P and w, which are action-angle variables for the unperturbed case (c = 0). Physically, these variables represent a polar coordinate

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system in the q-p plane where w is an angle and d2P/mw sin w,

q = d(2Pfmw) P = d2Pmw

With a simple canonical transformation terms of P and w as

where the prime denotes differentiation Hamilton’s equations for H* are

(2.3)

cos w.

[5] the Hamiltonian

l (w’/2w)P

H* = WP +

is a radial coordinate:

can be written in

sin 2w,

(2.4)

with respect to the variables 7 = ct.

ti = (aH*/aP)

= w + E(w’/~w)

P = (--aH*/aw)

= -E(w’/w)

sin 2w,

(2.5)

P cos 2w.

(2.6)

Equation (2.6) gives Vandervoort’s result for the change in the action integral due to the slow time variation of the Hamiltonian. According to Whiteman and McNamara [13] and Kruskal [l l] the first order adiabatic invariant for this problem is Z = P + ~(0’/2w~)P

sin 2~.

(2.7)

To verify the approximate constancy of Z, we take a time derivative of (2.7). Substituting for P and ti from (2.5) and (2.6) and keeping terms up to first order in E we obtain 1 = P + +J’/iw2)P

sin 2w + E(w’/w) P cos 2w + O(G).

The first and third terms on the right cancel by virtue of Vandervoort’s (2.6) leaving the lowest order

1’= l (co’/iw~)P

sin 2w + O(c2).

(2.8)

result (2.9

From (2.9) we see that, if (oJ’;~c.o~)= O(E) then f is of order c2 and hence Z is a first order invariant. This assumption is standard in asymptotic theories. It is equivalent to assuming a slow time dependence not only for the frequency w but also for the first derivative of the frequency w’ so that o’ = w’(d). A second order asymptotic calculation would similarly require that w” = u”(d) and so on. From the above development we see that the asymptotic result that Z is a first order adiabatic invariant is in no way inconsistent with Vandervoort’s first order iterative calcution of the change in the unperturbed action integral P. Rather, the two results are mutually dependent.

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We now consider the possibility of secular changes in the adiabatic invariant Z due to resonances between the oscillation and the time rate of change of the frequency w. We assume w of the form w = a, + 6 2

(2.10)

aned”+,

7L#Q

with w&J = O(E).

(2.11)

Using (2.10), f becomes to lowest order 1 =

2

($)”

z 0

c

n2an[ei(nwlt+2w)

_

eih,t-2w)],

(2.12)

TZ#O

where we have substituted Z for P to first order in E. A time average of (2.12) gives zero unless there is a commensurability between the oscillations in t and w, W/Wl = s/2,

(2.13)

where s is an integer of order (l/e). For the length of time that the system remains close to the resonance, there will be slowly varying terms in the sum of (2.12) which do not average to zero. These will be the n = fs terms. Assuming for simplicity that a-, = a, so that w is an even function of time, the average of (2.12) can be written as f/Z = (~/2)(.sw,/ao)za, sin 4, (2.14) where 4 is the slowly varying phase: + = so,t - 2w.

(2.15)

The a, generally decrease with increasing n, such that a, , itself may be of order (l/n)” = l where m depends on the differentiability of w. In this case the asymptotic series would not be expected to fail until the (m + 1)st order (see [6]). To find a new approximate invariant we employ the techniques of secular perturbation theory. We transform the action-angle variables for the unperturbed case (E = 0), P, w, to new variables .Z and 8 which are action-angle variables for the perturbed case (E # 0). The new variable .Z is just the adiabatic invariant Z defined in (2.7). The corresponding angle variable is e = w + E(W’/4W2) cos 2w,

(2.16)

H = WJ + +.IJ~/~~~).Jcos 28.

(2.17)

with Hamiltonian

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We now assume a resonance condition of the form (2.13) and transform to new canonical variables j and s which put the observer in a frame of reference rotating with the resonance. d is the slowly varying difference phase -4 of (2.15), and j is +&J. In terms ofj and d (2.17) becomes H = (2w - sq)J

+ E(W’j24j

cos(B + Swlt).

(2.18)

If the system remains close to resonance, 6 will be slowly varying compared to w,t and we expect that a time average of (2.18) would not appreciably effect the slow oscillation. Substituting --~s~w~~u~ cos sw,t for &‘, from (2.10), the average yields: R = (24 - SW&f - 42(s4aJ”

a,f cos B

(2.19)

which is independent of time and hence a constant of the motion. The equation defines j (or J) as an explicit function of 8. In the presence of the resonance, the simple adiabatic invariant J is no longer a constant to first order in E; instead there is a new approximate invariant defined by Eq. (2.19). For exact average resonance, such that the first term of (2.19) vanishes, there are infinities in J due to the independence of the average frequency from the action J. In actual systems nonlinearities limit the change in J. To see this, we assume that the average frequency a, depends nonlinearly on J in the following way: (2.20)

a, = a - ,8J,

where j3 = aa,/aJ, the first term in the Taylor expansion about J = 0. Substituting (2.20) into (2.19) we obtain, after some rearranging of terms,

H = swI

(

-1+2a-

__

-1

cos B

= Constant. We plotj

(2.21)

as a function

(2ol/sw,)

= 1.1,

of e”from (2.21) with, @/a) = .091,

(a&)

= .91,

and

E = .l.

For j = l/2, this corresponds to the case of average resonance in which ao/wl = (a - ~@)/QJ~ = s/2 so that the first term in (2.21) vanishes. The results (solid lines in Fig. 1) show closed phase loops centered about e^= fr. The areas within the closed phase loops are the adiabatic invariants of the transformed system, which replace the previous invariants that no longer exist in consequence of the resonant interaction.

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FIG. 1. Phase space of the oscillation in the adiabatic invariant due to resonance between the frequency of a one-dimensional oscillator and one harmonic of the time-varying frequency. The solid lines are plotted from (2.21) and the dashed lines from (2.23) with the constant chosen at the separatrix.

The results embodied functional form

in (2.21) can be made clearer by writing (2.21) in the

R = w,Y - 73 - d(Y) cos 19= const.

(2.22)

We note there is an elliptic singular point at J g l/2, 0 = -&r. This can be determined analytically by setting

aRjaf = aIQae^= 0. If we further expand j about the singularity,

as

Y=J+AJ and assuming (AJ/J) < 1, (2.22) yields Ye

+ d(J) cos d = const.

(2.23)

The separatrix for the oscillation is found from the Hamiltonian corresponding to the initial conditions AJ = 0 at d = 0, and the maximum value of AJ is then found at 6 = rr to be

(AJhn, = (2Gq)/y)1/2.

(2.24)

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We see that the maximum excursion of J due to the resonance is proportional to $12 or, more completely; proportional to the square root of the ratio of the coefficient of the perturbation term divided by the coefficient of the first order nonlinearity in J. This result, found by Rosenbluth et al. [19] for the perturbation of magnetic surfaces, is a general property of a system in which the nonlinearity occurs independently of the perturbation. If the nonlinearity occurs only in the perturbation then the island amplitude is governed by a different law, which we describe in the next section. We plot the curve of constant Hamiltonian at the separatrix, from the approximate expression given in (2.23), to obtain the dashed line in Fig. 1, which although differing in detail, gives a value of AJ,,, in good agreement with the results from (2.21). The frequency of the island oscillation can also be found near the elliptic singularity by linearizing (2.23) to obtain y(A J)” + &[(A @2/2] = cons&

(2.25)

with the frequency given by v = (2y41/2

which is also proportional to &J2. The new invariant generated by removing resonances can break down in two ways: (1) The frequency can drift away from the resonance adiabatically. (2) Neighboring resonances can interact sufficiently strongly that a single resonance does not describe the motion. In the first case an adiabatic invariant usually exists, but the phase curves are distorted by the passage of frequencies near successive resonances. In the case of strongly interacting resonances, values of E can be chosen appropriately to give both succesive island resonances and ergodic regions between them. For larger perturbations the islands can interact sufficiently strongly for the entire island structure to be absorbed by the ergodic region. The width of a resonance relative to the distance between resonances is the important parameter in determining breakdown of invariants. We have obtained an approximate formula for the width of the s-resonance in terms of the variation of the action. We obtain the variation of the frequency from Aw = (dw/dcf) 2AJ

and compare it with the separation between the s and s + 1 resonance which is approximately w/s, to obtain a criterion for resonance overlap (s/w)(dw/dj)

2AJ = O(1)

(2.27)

and since da/d3 G +y, we have, using (2.24) and (2.26), that for overlap sv/w = O(1).

(2.28)

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We explore the validity of an overlap criterion in the detailed numerical example of Section IV. There, however, the nonlinearity appears in the slow oscillation which modifies the criterion as given in (2.28).

III. A.

GENERAL

THEORY

Removal of Degeneracies

We assumethe Hamiltonian consists of two parts in the form H = H,, + eHI , where H,, is solvable in action angle variables such that

H = HoV’, , f’z>+ l Hdp, , f’z , WI, w,>,

(3.1)

with Ea small number, and HI is assumedto be periodic in w1 and w2with period 27r such that it can be expanded as HI = 1 H!,,JP, , Pz) ei(ew1+mw2). 8.m If a resonance exists between the unperturbed frequencies,

b2Ypl , p2)IvlYpl , &)I = 4s

(r, s integers),

(3.2)

where VI0 = aH,/aP, ;

vo2= aHojap,,

then an attempt to solve the motion in action-angle variables by perturbation theory leads to a secularity in the solution 1251.We will take Eq. (3.2) to represent either a primary resonance in the system or a secondary resonance created by harmonic frequencies of phaseloops generated by the primary resonance. In either case,the secularity can be removed by applying a transformation which eliminates one of the original actions, PI or P, , from the unperturbed Hamiltonian Ho . We choose the generating function F2 = (rwl - SWJPI + w,p2

(3.3)

which defines a canonical transformation from PI , Pz , w1, w2 to p1 , pS , zC1, z& such that

ti, = (aF2/aPl) = rwl dir,= (aF,laP2) = w2,

SW~,

Pl = (aF,/aw,) = rPI, P, = (aF2/aw,) = li, - s&.

(3.4)

These coordinates put the observer in a rotating frame in which the rate of change of the new variable z& = rti, - sti, measuresthe slow deviation from resonance.

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In removing a degeneracy by use of a generating function as in (3.3), there is an arbitrary choice of which of the original phase variables to leave unchanged. We assume here that w2 is the slower of the two phase variables and we leave w2 = zZ, unchanged such that in averaging the Hamiltonian over the fast variable after the transformation the average is taken over the slower of the original variables. This choice is necessary if higher order resonances are to be removed, in order to retain the lowest harmonic of the second order interaction. The averages over the second order resonances are more conveniently performed on the faster variable. Provided resonances higher than second order are not important, this procedure is equivalent to the average over the slower variable. We consider two cases: (1) If the resonance condition is met for the unperturbed frequencies for all PI and P, , then the Hamiltonian is intrinsically degenerate. (2) If the resonance is satisfied only for a particular value of P, and P2 , then the Hamiltonian is accidentally degenerate. A primary resonance can be either accidental or intrinsic, but a secondary resonance is almost always accidental due to the complicated way in which the frequencies depend on the actions. Applying the transformation (3.4) to the Hamiltonian, we obtain

for the intrinsically

degenerate case, and

for the accidentally

degenerate case, where

Although (3.6) has the same form as (3.1), & < & allowing an average over zZz , as we will show below. In the case of intrinsic degeneracy [Eq. (3.5)], Hamiltonian’s equations are

715~ = aHlaP = c(aA/aPl) = o(+,

(3.7)

S, = aHlaP = a*laP2 + c(anlaP2) = o(i),

(3.8)

where we have assumed ZUl,,,/a~l = O(1) in (3.7). We see that &, is slowly varying compared to z& and hence we can average (3.5) over ti, to give

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where

A = & 1’” A(Pl ) P2, zzl) 252) 0

d4,

(3.10)

where we have assumed A-!,, = .L$,- as a reality condition independent of ti2 , we have the approximate result that

on fl. Since R is

P2 N const.

(3.11)

This is the first term of the series for the adiabatic invariant of Hamiltonian (3.5). We see from (3.4) that p2 represents a combined invariant for the degenerate system, namely p2 = P2 + (s/r) PI . The effect of the rotating coordinates is to explicitly exhibit the single invariant of the resonant system. Notice, however, that for a high order resonance in which I <
=

(3.9) can be expanded about the singularity @2>

+

4fL

Jj2,

a + aw)2/21

+fKw2/4

as

+ .*.,

(3.13)

where AP=fj-&, Aw = ti, - 6,) and

(3.14)

where we have assumed a2Ao,o/aP12 = O(1). The coefficient of the AP Aw cross term has been assumed to vanish in the average since this is the observed behavior in our numerical examples. Iff, g > 0, the singular point is elliptic and Hamiltonian

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(3.13) can be used to plot elliptic phase orbits in the AP-dw phase plane. The frequency of oscillation around a phase loop in general depends on the area enclosed by the loop; however, for loops close to the singularity, this frequency approaches a constant value of [17] coo = (J’g)“” = O(dlyJ, i.e., the frequency of the PI - 6, oscillation is a factor of •4?:,~ slower than the frequency of the p2 - ~2, oscillation. The ratio of the lengths of the semiaxes of the ellipse can be calculated approximately as [ 171

such that if the maximum excursion in dw is of order unity, then the maximum excursion in APjrj is of order L’FF,~. To complete the solution, formally, we transform to action-angle variables for the slow oscillation of AP and Aw. We postpone this calculation until we have considered the more general case of accidental degeneracy. The transformation to action angle variables is generally unnecessary unless a secondary resonance must also be removed. In the case of accidental degeneracy, we consider the Hamiltonian of Eq. (3.6). The unperturbed part of the Hamiltonian in (3.6) is not independent of p, as for intrinsic degeneracy. Hence, it is not obvious that any averaging can be performed. However, because the hat variables explicitly exhibit a singularity in the PI - tiI phase plane, an average may be performed in a region of the phase plane sufficiently close to the singularity. Performing the average on (3.6) JJ = $48 >6) + 4Pl

, p2,q

(3.15)

which, when expanded about the singularity, gives

17=~(I’,,~2)+~ii(~l,~2,Ptl)+C(~+F~+...,

(3.16)

where (3.17) (3.18) The linear frequency of the PI - zZ, motion is then 80 = (jql’2 595/7+3

= O(EqpQ

(3.19)

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which is a factor of •~/~L’P/~ -~,s slower than the frequency of the p, - ti, oscillation, thus justifying the average. The ratio of the semiaxes of the ellipse is R = (dP),,@&,,

= (F/c)“”

= O(E~‘~~$~).

(3.20)

Pz is an approximate constant of the motion, and, together with (3.15) isolates the Pl - &I phase trajectories. In the example of Section IV, the Fourier coefficient, /1-,,, , is of order unity for the primary resonance. This leads to values of sZ”, and R in Eqs. (3.19) and (3.20) which are of order &j2. For the secondary resonance, the Fourier coefficient corresponding to II-,,, is strongly related to E giving dependences of sZ” and R which depend on E more strongly that Gi2. An accidental degeneracy in a multidimensional system is equivalent to a resonance of the one dimensional anharmonic oscillator with periodically varying frequency, which we treated in Section II. In both cases the frequency is a function of the momentum in the absence of the resonant coupling. B.

Higher Order Resonances

If E is not sufficiently small, secondary resonances which are present in Hamiltonians (3.5) and (3.6) contribute secular terms that modify the invariant pz . These resonances are between harmonic frequencies of the p1 - eir, phase oscillation derived in Section A, and the fundamental frequency v2 . They can be removed in a manner closely analogous to that used in Section A. However, the results have some additional features which will be seen by carrying through some of the steps explicitly. In order to apply the theory of Section A, the part of the Hamiltonian not containing the resonances must be in action-angle form. We transform to action-angle variables fi , & by solving the Hamilton-Jacobi equation, (3.21) RPl 9 p2 9 C) = ml 3 P2,>? where, for an accidental degeneracy, we substitute in Eq. (3.16) for i!i. We obtain a solution near the elliptic singularity from perturbation theory. Keeping only quadratic terms in dP and dw, the transformation to action-angle variables JIO, 010 for the harmonic oscillator yields [5]

The old and new variables are related by the generating function -Fl = &(F/G)l12 (Aw)~ cot e”

MODIFICATION

335

OF INVARIANTS

such that AP = (2J,“R)1/2 cos 8,O,

(3.23)

Aw = (2J10/R)1’2 sin fIIo.

The nonlinearity is reintroduced by keeping terms up to fourth order in AP and Aw, and transforming to new action-angle variables J1 , ~9~using perturbation theory:

where B, = @JO17

(3.24)

and R,/Ro = O(h),

R,/R”

= O(P)

with h = (2J,“R)312/2QoJlo.

(3.25)

Recalling that JImax = ~(AP)ma,(Aw)max = &R(Aw)&~~, in the accidental case J lmax = O(E~/~A!!~,~) and A,,, = O(E~/~LI?‘~.~). Therefore j B, 1 Q 1R, 1 < j W, 1 and we can write ii; as

R = *@?> P2) + 4&,

&,61) + 17, + 6 * R, + 62 . R, ,

(3.26)

where 6 is an artificial constant measure of smallness which is set equal to unity at the end of the calculation. For the intrinsic case, however, JImax = O(A?F,,) so that A,,, = O(A~~,s/~) and h therefore is not necessarily small. Hence the expansion (3.26) will have a much greater range of validity in the accidental case than in the intrinsic case. To apply perturbation theory, we make the following expansions for the new Hamiltonian and the generating function K = K,, + SK, + a2K2, s

=

so

+

ss,

+

s2s2

)

where S, is the identity transformation, f310J,. We then solve for K and S to each order in 6 (See [25]). The results are, to first order,

and

Q,“(J&%‘,la40> = -[K(4°,

JI) - &
(3.27)

336

JAEGER

AND

LICHTENBERG

and to second order

as2 ,el”h K2 =Qvl) aB,o +z 13lo+~2Vl

where E1 is the average of H, over 01o. Since R, has only sinusoids in elO, we have B1 = 0. Thus there is no frequency and we must evaluate K, to obtain the lowest order effect of the frequency shift. This is accomplished by averaging K, over

(3.28)

odd powers of the shift to first order, the nonlinearity on 19~0giving (3.29)

If no first order terms appear in the Hamiltonian, Hr = 0, and K, can be obtained from the simple average & . Substituting for dP and.dw in terms of J1 and 8, yields for the average part of the Hamiltonian

where m is in general a complicated function of pz . Equation (3.30) is the formal solution in the case that the average over 6, is valid. It is independent of angles so that pz and J1 are the two constants of the motion. To take into account the effect of the secondary resonance in modifying this solution, we reintroduce the terms ignored in the simple average over &ir, . ml

3 p2 2 4 3 62) = a&,

p2 , dl , fi,) - 4c

3p2 ,a

(3.31)

with Fourier expansion directly in hat variables (1” =

Expanding

(1” =

+ m&J].

(3.32)

+ dw] + m&)1.

(3.33)

about the elliptic singularity gives A =

Transforming

C Ae,m(Pl , P2) exp[i(&, e.m m#O

C At.,
+ dP, P2) exp[i@J

to lowest order action angle variables by (3.25) , C 4.APl

e.m. rn#O

, p2,> exp[i(&

+ m&>] exp [if (+)l’l

sin

e,]. (3.34)

MODIFICATION

OF

337

INVARIANTS

Here, we have kept only the first term in the expansion for d w. We neglect the next term, of order h = (E~/~LI?~,,) and the lowest order term in AP = O(E~/~A?~,~). Expanding the second exponential in (3.34), and dropping the hats for convenience, we obtain

if = 1 rn,,(Jl, P2) expMn4 + mw2>l m.n rn#O

(3.35)

where r,,,(J,

, P2) = c eieE1&,(P, c

, Pz) J,

(3.36)

and J,, is the Bessel function of order n. From (3.35), it is evident that there can be higher order resonances between w2 and O1such that the average of (3.36) over w2 is not zero. To obtain the new invariant, we write the total Hamiltonian as

H = K(J, , Pz>+ ~(1”<5, , f’, ,h , WA

(3.37)

which has a form similar to (3.1) so that we can use the method of Section A to remove the secondary resonance. Assuming a resonance v2l-Q

= PI4

(3.38)

CP, 4 integers),

where

'2 = -ap, = O(l), a = g

= o(2'2Ay?s), 1

we transform to new variables j, , p2 , 8, , ti, as in (3.3) where 8, is the slow variable pB1 - qw, , derived from the generating function, r;, = (Pb -

w2)

31 +

w2p2

*

An average over the fast zZ2is then equivalent to-keeping only m = q, n = -p, and m = -9, n = p, together with harmonics, in /I. We obtain

(1; = &

j-‘” d dzi2 = 2 f 0

t=1

r-t,,&

, &) ~0s tb .

338

JAEGER

AND

LICHTENBERG

The double bar distinguisges this average from the average of Section A which ignores all of the terms in L’L Since R is cyclic in ti, , we have also P2 = P, + (q/p) J1 11 const

(3.41)

such that the jl - 8, phase orbits are isolated. Comparing equations (3.39) and (3.40) to equations (3.15) and (3.10), we see that all the results of section A for accidental degeneracies apply to the higher order resonance of this section if we replace the Fourier coefficient L,,, with the coefficient .l& . From (3.36) we see that the largest term in r--9,9 is proportional to J,(d2J,/R), wherep is a large integer of order (qc-1/2A:~!~). Since .I1 M8x is proportional to R, the maximum value of the argument (2J,/R)‘l” is of order unity which allows the Bessel function to be approximated by the first term in the expansion for small argument,

J,

d[

1

. ! 1*

(qE-w~~~7’4)

(3.42)

From (3.42) we also see that the amplitude of the interaction term is proportional to Jt’” such that the island oscillations decrease in size rapidly with decreasing J1 . Replacing A-,,, with C,,, in Eqs. (3.19) and (3.20), and using (3.42) we see that the amplitude and frequency of the oscillation in jl will be at least a factor of O(l/~-l/~!)l/~ smaller than the amplitude and frequency of the oscillation in PI . We call the new jl oscillation an island oscillation because it appears as a chain of islands in the P, - w1 phase plane. Although the procedure for exhibiting the invariant curves of the island oscillation, arising from the second order resonance, is the same as that used for obtaining the invariant curves of the primary resonance, the results have a somewhat different character. The strength of the island resonance is related explicitly to a particular form of phase nonlinearity, depending strongly on E. The strength of the primary resonance is related weakly to E, as l 1j2. Thus, for relatively small E, island oscillations rapidly become of negligible importance. For relatively large E, on the other hand, the island oscillation may be more important than the primary resonance in determining the limits of adiabatic invariance. In principle, the above procedure may be extended to third order resonances, but because of the factorial dependence on E we find that the island oscillations either interact sufficiently strongly to terminate the series or the third order resonances have vanishingly small effect. For intrinsic degeneracies the second order resonances do not have the same explicit form of E dependence. However, the conclusion above, that the importance of the island oscillations is strongly dependent on the magnitude of E, is still valid.

MODIFICATION

IV. NONLINEAR

339

OF INVARIANTS COUPLED OSCILLATOR

In this section, we present results from an example which has an intrinsically degenerate fundamental resonance. By introducing a term to make the degeneracy accidental, we can compare the two problems and observe numerically the differenceswhich we predicted analytically in the last section. We consider a two-dimensional oscillator in x and y with the Hamiltonian H = Ho(P, 3Pv 2 x, u) + 4(X,

.Y>,

(4.1)

where Ho = pn2/2 + x2/2 + py2/2 + 2y2,

(4.2)

HI = x”y - (4/3) y3.

(4.3)

and E is an artificial small parameter to be set equal to unity at the end of the calculation. The significance of Eis to remind us that for small amplitude oscillations in x and y, the terms in the perturbation HI are smaller than the terms in Ho by a factor of the square root of the unperturbed energy. The potential, U(x, y), is U(x, y) = $(x2 + 4y2 + 2xzy - fy”). As U increases, the curves of equipotential (U = constant) become increasingly nonlinear until a triangular separatrix at U = .6666 is reached, which separates bound from unbound trajectories. We transform (4.1) to the appropriate form to apply our general theory by introducing in two dimensions, the action-angle variables for the harmonic oscillator as given in (3.26) x = 2/‘2p, sin 1)~~ pa = d2P, cos M’2

y = x@ sin w1

(4.4)

py = 2/4p, cos WI,

which gives H = Ho(P, , f’2)

+ 4(P,

2f’2,

~1,

w,),

(4.5)

where Ho =

P2

+

=‘I,

(4.6)

and Hl = 2P2(Pl)‘lz sin2wz sin w1 - +(P,)3/2 sin3wl. The unperturbed frequencies in the two degreesof freedom are then lYIJzo = aHo/aP2 = 1,

wyo = aH,Iap, = 2

(4.7)

340

JAEGER

AND

LICHTENBERG

so that the resonance condition becomes wyywzo = 2.

(4.8)

Note that this is intrinsic rather than accidental because wZo and wyo are independent of P, and Pz . This problem bears close resemblence to the problem treated by HCnon and Heiles [ 151 and McNamara and Whiteman [ 121. The only difference is that we have assumed a one-to-two resonance between the frequencies wZo and wyo while the above authors treat the one to one resonance. The advantage of the one to two case is that it provides a lowest order invariant which is zero order rather than first order in E, and hence it is easier to calculate. To remove the intrinsic degeneracy, we employ the generating functions as in (3.3) Fz = (wl - 2w,) PI + w,pp

giving a transformation part of the Hamiltonian

to new variables PI , PZ ,8, , ti, such that the unperturbed Ho is a function of P, alone, Ho = & .

The total Hamiltonian,

(4.9)

(4.10)

in terms of the hat variables, is

H = r”, + 2(1”, - 2&) p;”

sin2 eir, sin(ti, + 26,) - @f’” sin3(8, + 28,),

(4.11)

and can be averaged over ti, to give i7 = P2 - i(P2 - 2PI) Pf12 sin ti, ,

(4.12)

where we have set E = 1. The lowest order adiabatic invariant is therefore, P2 = P2 + 2P, = const

(4.13)

which, together with H = constant, isolates the PI - dI phase motion. The light lines in Fig. 2 show the PI - tiI phase loops plotted from equation (4.12) for p2 = .08333 and a total energy E = .08333. As a check on the averaging procedure, we have also shown in Fig. 2 (heavy lines) the trajectories obtained by numerically integrating Hamilton’s equations before the average over eir, . We differentiate equation (4.11) with respect to PI , p2 , tiI , ti, and solve the resulting four simultaneous differential equations by a standard subroutine on CDC 6400 computer. The numbers in Fig. 2 represent successive crossings of a plane of section in x: (4.14) sin 27G2= 1.0.

MODIFICATION

0.04

OF

INVARIANTS

E

q

341

0.08333

0.03

6, 0.02,-

0.0 I-

( FIG. 2. Phase trajectories resulting from removal of an intrinsic degeneracy in a two-dimensional coupled oscillator. The dark lines show the Hamiltonian curves before averaging over the faster of the two oscillations and the light lines show the Hamiltonian curves after averaging.

There are approximately five oscillations in x for each oscillation in PI , the trajectory crossing the plane of section twice during each oscillation in x. The slight scatter in the points plotted is due to inaccuracies in satisfying (4.14) exactly. The relatively smooth curves traced out in the PI - 6, phase plane indicate the existence of an adiabatic invariant which in this case we know is pS to lowest order. Comparing the analytic and numerical solutions in Fig. 2, we seethat there is slightly more asymmetry in the numerical result. Presumably, if we carry the averaging procedure to higher orders in E, the agreement will improve. Also, from Fig. 2, we notice that for the outer-most phase loops, the fractional change in 8, is of order unity, and the fractional change in PI is likewise of order unity. This result is a direct consequenceof the intrinsic nature of the resonance as we showed in the last section.

342

JAEGER AND LICHTENBERG

Alternaely, we can follow McNamara and Whiteman [12, 131 by writing the first order adiabatic invariant Z in terms of x, y, pz , pr , eliminating pz between Z = constant and H = E, a constant, and evaluating the result in the x = 0 plane of section. We obtain, E + iy(2E + $y3 - py2 - 4y2) = const

(4.15)

which gives the isolated motion in the y - py phase plane shown by the light lines in Fig. 3. Both McNamara and Whiteman [12] and H&non and Heiles [15] show 0.4

1

0.2 PY 0.0

-0.2

-0.4

1

,

-0.2

I

-0. I

/

t

j

0

0. I

0.2

Y FIG. 3. Phase trajectories corresponding of the y oscillation.

to those in Fig. 2, but plotted in the phase plane

results in this plane rather than in the PI - z& plane of Fig. 2. The two methods of plotting are equivalent, but that in Fig. 2 is in a more convenient form for subsequent transformations. The dark lines in Fig. 3 show numerical results obtained as in Fig. 2 and plotted in the plane of section x = 0. Again there is more asymmetry in the numerical result, but as McNamara and Whiteman [12] have shown, if the adiabatic invariant is calculated to higher order, the agreement can be quite good. For the oscillator with w,O = my0= 1, they obtain reasonable agreement in fourth order for this sameenergy. This is not the case, however, at higher values of the energy for which higher

MODIFICATION

343

OF INVARIANTS

order resonanceshave significant amplitude. This is shown by the island formation in Fig. 4 which was calculated numerically for an energy of E = .33333. The noncornected points correspond to a single particle orbit as do the island trajectories. 5

0.2501p

-77

T/2

FIG. 4. Breakup of the Hamiltonian when the energy is increased.

0

E=0.33333

K/2

7r

curves before averaging due to higher order resonances

A corresponding curve plotted from Eq. (4.12) shows neither the ergodic region nor the chain of islands sincethese features arise from the second order resonances which are ignored in the averaging leading to (4.12). The frequency of the phase oscillation in Fig. 4, as seen from the numbering of the crossings of the surface of section, is approximately twice that in Fig. 2. Since the energy is 4 times as large, this confirms the general result that phase oscillations resulting from the removal of an intrinsic degeneracy vary as E = VE and it accounts for the relatively high value of energy (E = .3333 in Fig. 4) necessaryto observe breakdown. To contrast this behavior with that resulting from the removal of an accidental degeneracy, we introduce another term in the unperturbed Hamiltonian (4.6)

344

JAEGER

AND

LICHTENBERG .

frequencies wIo and wzo depend on P, and P, . We take

so that the unperturbed

(4.16)

Ho = P, f 2P, - P;‘“P,““,

giving w,o = aH,laP,

= 1-

&(PI/Pp2

w,o = aH,Iap,

= 2 -

*(P21Pp.

0.401

E = 0.0068

Pz= 0.01051

(4.17)

&+P,) I

0.35 102 0.30-

0.25-

0.20.

(a) 0.40

E = 0.0068

curves resulting from removal of an accidental degeneracy FIG. 5. (a) Averaged Hamiltonian in a twodimensional coupled oscillator. (b) Hamiltonian curves before averaging for parameters corresponding to those in (a).

MODIFICATION

OF

345

INVARIANT’S

Although the additional term in Ho has been introduced somewhat artificially here, such terms do occur naturally in many problems. An example of such a problem is cyclotron resonance between a particle gyrating in a magnetic mirror field and an electromagnetic wave [27]. With the nonlinear term present in the unperturbed Hamiltonian, the transformation (4.9) doesnot eliminate the pr dependencefrom Ho. But as we saw in Section III, an average over 6, is still justified close to the elliptic singularity in the pr - ti, phaseplane. Performing the average we obtain

with a lowest order invariant pz as defined in (4.13). In Fig. 5a, we show the pr - &r phase curves plotted from (4.18) for p’z = .01051 and a total energy of E = .0068. The outer most phase loop shows a maximum fractional change in 6, of order unity while the maximum fractional change in p1 is 0.3 which is of the order of the fourth root of the energy, consistent with the general results given in Section III for accidental degeneracies.The corresponding numerical plot, before the average over eir,, is shown in Fig. 5b. Becausethe frequency of the p1 - 8, phase oscillation in the accidental case varies as Gjz E E114 whereas it varies as E E E112 in the intrinsic case, a much smaller value of the energy is required for accidental degeneracy to obtain a given resonance. The ratio of the frequency of the five-island oscillation indicated by the sets of numbers forming the five small ellipses, to the frequency of the main invariant curves is observed to be l/6 which is in reasonable agreement with the predicted value of (l/p!)li2 z l/l1 from Section III. A plot of pz averaged over the oscillation for the five-island trajectory is shown 0.1065

.

.

&IO .

0.1060I]

,

.

.

l l

..

.

l*

.

. l

.

.

.

. . . . .

.. ..

. ..

0.1055-

Number of crossing

FIG. 6. Plot of the lowest order invariant 4 averaged over the period of the oscillation in x, for the five-island trajectory in Fig. 5b.

346

JAEGER

AND

LICHTENBERG

in Fig. 6. pZ oscillates with the period of the island oscillation as is shown by the numbering of the points. Points l-11 lie on the outside of the islands in Fig. 5b and enclose a large area inside their phase loop. We see from Fig. 6 that these same points have relatively low values of pZ . On the other hand, points 31-41 lie on the inside of the islands enclosing a smaller area inside their phase orbit, and we see from Fig. 6 that these points have relatively high values of pZ . In Fig. 5b, the isolated points, outside of the closed curves, all belong to a single ergodic trajectory. The points on the border of the ergodic region which are connected by solid and dashed lines show the partial, unstable formation of chains of six and seven islands, respectively, indicating a strong interaction or “overlap” of these chains of islands. The stability of the five-island trajectory which is closer to the elliptic singularity is to be expected from the results of Section III where we saw that the secular terms have amplitudes depending on Bessel functions with arguments proportional to the area enclosed by p1 - til phase loops. Hence for the five-island trajectory, the area enclosed is less than for the six-and seven-island cases and the secular terms become less important. The index of the Bessel function, on the other hand, depends on the resonance number so that the size of the islands decreases as the number of islands goes up. These competing effects make the sizes of the five and six island chains in Fig. 5b comparable. The higher harmonic island chains are also closertogether, increasing the interaction of neighboring island chains. If the energy is chosen somewhat lower it is possible to obtain breakup of the six island chain without direct overlap of the neighboring chain of seven islands. In this case the nonresonant terms perturb the motion sufficiently that the island stability is destroyed, although the island may hold its general shape for a number of periods. We note in Fig. 5b that an 11 island chain also appears, corresponding to the 11/4 resonance, such that the trajectory returns to a given island after two oscillations of the main resonance. The islands are of much smaller amplitude as the interaction strength depends on the amplitude of a Bessel function of 1Ith order, rather than the 5th and 6th order Bessel functions corresponding to the 5 and 6 island trajectories, Narrow bands of stochasticity may exist near the separatrices of the 5 and 11 island chain resonances, but these trajectories are enclosed by adiabatic trajectories which confine them to limited regions of the phase plane. The main invariant curves in Fig. 5b represent oscillations of the double oscillator system about the fundamental resonance, wyo/woo = 2. The question naturally arises as to the nature of neighboring resonant oscillations such as that about wVo/wZo = 3. In this case, eir, is no longer slowly varying but rather, && = WY0 - 2w,o N w z0. This means that, for each cycle of the x oscillation,

the system passes through one

MODIFICATION

OF INVARIANTS

347

complete oscillation in the PI - dir, phase plane. This leads to the two narrow islands at the top of Fig. 5b. These islands are much smaller than the main 2 to 1 resonance phase curves because the resonance condition wIo/wZo = 3 can only be satisfied in a region of strong nonlinearity as opposed to the CIQ~/CIJ,~= 2 condition which is true in the entire linear region. To analytically treat the higher order resonance between the rapid oscillation in x and the slow drift in PI - tir , we transform to the action-angle variables of the $ oscillation. As in Section III we expand the variables PI , dir, , about the elliptic singularity, as P, = &+

AP,

Gl = & + Aw

(4.19)

such that transformations (3.25) can be used to obtain the action-angle variables for the linearized motion about the singular point. Applying perturbation theory as in Section III we obtain the averaged Hamiltonian in action angle variables of the form R = P, - (p2 - 2191’2 p;‘2 + F(p2) + Q”(~2) Jl[l + A2(p2 , J1) M(p2)],

(4.20)

where

Q”, is the linear frequency of the PI - Z& phase oscillation, and X = LX&~. F, R, sZ” and M are functions of p2 resulting from the expansion which we derive in Appendix 1. The varying part of the Hamiltonian can be Fourier analyzed as [(2J,R)3/3/2QOJ,]

Q=H-jj = ; [(P2 -

2P,) By

-

(4.21) where we have omitted the hats here in order that we can introduce a second set of hat variables to remove the higher order resonance. The total Hamiltonian is H =

R(f’2

>4) +

fiV2

>4 , ~2 , 4)

(4.22)

where B and A are given by (4.20) and (4.21) respectively. We define the unperturbed frequencies: v2(p2 , .rl) =

aR/aP, ,

(4.23)

348

JAEGER

the oscillation

AND

LICHTENBERG

frequency in the x - pz phase plane, and LqP2 ) J1) = aR/aJ, )

(4.24)

the oscillation frequency in the P, - wr phase plane. We see from the exponent in (4.21) that there can be resonances in (4.22) of the form c2(P2 , JlSV2

(r is an integer).

, J1) = 42

(4.25)

These resonances will introduce secularities in the time rate of change of the adiabatic invariant Pz . The resonances only occur for certain values of J1 and P, which, themselves vary due to the secularities. As we see in Fig. 5b, islands form around elliptic singularities in the J1 , O1plane. Hence we can transform to a second set of hat variables, pz , jl , ZL&, 8, , in which 8, is the difference phase rL$ - 2w, which is slowly varying near the singularity. The required generating function is F2 = (4 - 2w,)j, which defines the transformation

(4.26)

+ w2 , p2

to the hat variables in the rotating frame,

ti, = aF2/iYp2 = w2

P, = aF,/aw,

8, = aF21ajl = re, - 2w,

J1 = aqae, = rjl .

Writing the perturbation we get 6 = [(p2 -

= -2y,

+ P2 (4.27)

(4.21) in terms of the hat variables and averaging over ti2 ,

231 - 2P,) ,;”

- Pf’2] J,.

- ; (rj, - 21, - 2a,) P;‘“J,, (4.28)

The double bar average corresponds to keeping just the most slowly varying terms of the sum (4.21) which, for the C,/Q = r/2 resonance, are the L = fr, f2r, &3r terms. Since the 6’ = f2r, *3r terms are considerably smaller than the L = &r term due to the Bessel function weighting factors, we drop the last two terms in (4.28) and write the approximate average Hamiltonian as R = p2 - 2y, -

(p2 - 2j, - 2&)1’2 P;‘2 + F(p2 , 1,)

+ Q”P2 , h) r3Jl

+ h2P2 , A) MP2

+ [(p2 - 2y, - 2P,) ,;”

- P;‘“] J,

, LPI (4.29)

MODIFICATION

OF

349

INVARIANTS

where we have taken into account both whole integer and half integer higher order resonances between the x oscillation and the slow drift in PI - w1 . Since i7 is independent of &, , the adiabatic invariant for the island case is P2 = P, + (2/r) J1 E const.

(4.30)

In Fig. 7, we plot the j1 , 8, phase trajectories from (4.29) for various values of the “2 i-l

LO-

0 -lT FIG.

7. Transformed

I -a/2

/ 0

A 4

I T/2

7T

phase space of the five-island oscillation shown in Fig. 5b.

constant R. We have chosen r = 5 corresponding to the five-island trajectory m Fig. 5b. On the right hand axis we give the ratio vJ.Q. The oscillation centers about an average bounce-energy resonance number of Q/L? = 2.5 as does the five-island trajectory in the numerical integration. At the extremes of the oscillation, the ratio v,lQ never moves very far from 2.5 going to 2.54 at the top of the phase loop and 2.47 at the bottom, thus justifying the fundamental assumption, for keeping only the most slowly varying terms, that the system should remain close to a particular higher order resonance. AIso, we see, from (4.30), that if J1 oscillates then P2 , the adiabatic invariant without resonances must also oscillate, explaining the synchronized oscillation between P, and J1 observed in Figs. 5b and 6. Given the oscillation in j, = (I/r) J1 from any one of the curves in Fig. 7, we can calculate the amplitude of the resulting oscillation in the first order invariant P, from (4.30) with r = 5. For the outermost closed loop in Fig. 7, we obtain

~@Pz)mitx/Pzl =

.40 %.

From Fig. 6, the relative change in P, given by the numerical integration is [(dPz)max/Pz] N .47 ‘& slightly larger than the analytical result. An under595/7+4

350

JAEGER

AND

LICHTENBERG

estimation of the variation of Pz is to be expected, analytically, since we have overestimated the nonlinearity by expanding the average Hamiltonian to only fourth order in AP and Aw. The next term would be of opposite sign thus reducing the nonlinearity and making the amplitude of the predicted island oscillations larger. In the numerical integrations of Fig. 5b, we also find relatively smooth phase loops near the elliptic singular point and ergodic trajectories near the separatrix. By plotting f, , 8, phase diagrams for these other types of behavior, we can distinguish the physical mechanism that differentiates among them. Taking r = 6 and r = 7 leads to additional oscillations about u,/L? = 3.0 and vJSZ = 3.5 which are similar to those of Fig. 7. Also, r = 4 contributes nonresonant drifting oscillations. Rather than plot these additional trajectories in Fig. 7, we have transformed from the 4, - 8, variables back to the p1 - ti, variables of Figs. 5a and 5b. Fig. 8 shows the resulting curves for r = 4, r = 5, r = 6, and r = 7. The chains of five, six, and seven islands correspond to r = 5, 6, and 7, respectively, and the nonresonant drifting curves are shown as dashed lines corresponding to r = 4. The 2.5 resonance oscillation is fairly well separated from neighboring oscillations on either side, whereas the 6 island and 7 island oscillations are

0.35 1 0.300.250.20 -

FIG. 8. Averaged Hamiltonian curves resulting from removal of the five-, six-, and sevenisland resonances. Drifting curves resulting from the four-island resonance are shown as dashed lines.

MODIFICATION

OF INVARIANTS

3.51

relatively closer together. Undoubtedly, this accounts for the stability of the five islands observed numerically in Fig. 5b, and the instability of the chains of six and seven islands. The effect of the nonresonant, drifting terms caused by the very low order resonances (r = 4) is not entirely clear. These terms are very much smaller near the five island trajectory than near the six-and seven-island trajectories. This is to be expected, as the strength of the harmonic terms is proportional to Bessel functions depending on J1“’ . One would expect the nonresonant terms to average nearly to zero close to the island’s singularity. To see this, we recall that the nonresonant terms average exactly to zero when the system is exactly at resonance due to the orthogonality of the exponentials making up the Fourier series (4.21). If the system is not exactly at resonance, the nonresonant terms do contribute, but in the approximation of a symmetric oscillation about resonance, the contributions above and below resonance cancel. The assumption of symmetry around the resonance is valid in the linear region close to the island singularity so that the total nonresonant contribution averages nearly to zero over the island oscillation. If neighboring island oscillation are close together, however, as are the six and seven island oscillation in Fig. 8, the nonresonant terms do not average to zero. For example, they perturb the orbits calculated from the resonant terms which generate the six-island trajectory causing the orbit to drift to the seven island trajectory before completing the six-island oscillation. In this way, the contributions of the drifting terms above and below resonance do not cancel. This. accounts for the apparent “overlap” of the six-and seven-island oscillations observed in Fig. 5b. V. CONCLUSIONS For multidimensional oscillatory systems with widely differing periods, adiabatic invariants can be found that separate the degrees of freedom. If a low harmonic number resonance exists between two degrees of freedom a transformation may be employed to remove the resonance. Two cases must be distinguished: (1) Intrinsic degeneracy in which the frequencies of the oscillations are independent of the momenta in the absence of the terms which couple the two degrees of freedom. In this case the transformed degrees of freedom have frequencies that are separated by the strength of the coupling term E. (2) Accidental degeneracy in which the oscillations depend on the momenta. In this case the transformed degrees of freedom have frequencies whose ratio is proportional to ~‘1~. In either case, if E is small enough, adiabatic invariants exist in the transformed coordinates that separate the degrees of freedom. For increasingly large E, harmonics of the slow oscillation resonate with the fast oscillation to perturb the invariants. The harmonic amplitudes are strongly dependent on the harmonic number, and therefore the lower harmonic resonances of an accidental degeneracy will become important

352

JAEGER

AND

LICHTENBERG

at a lower value of E than for an intrinsic degeneracy. The perturbation of the invariants leads to islands which have new invariants associated with them. The island invariant is obtained by removal of the island resonance. The procedure of removing resonances can be carried to higher order, by considering resonances between the island oscillations and the faster periods. The frequencies of the island oscillations are slower than the next faster frequency by O(l/~-l/~!)l/~. In practical cases the strength of the harmonic amplitudes contributing to the island formation decreases rapidly such that higher order resonances need not be considered. There is a rapid transition with increasing E between the value at which significant islands appear and the somewhat larger value of E at which the terms contributing to neighboring islands interact sufficiently strongly to destroy the invariants of the motion. Invariant destruction leads to apparent ergodic filling of the phase plane. From numerical computations the invariants are first destroyed in parts of the phase plane where the island interaction perturbs the invariants such that the amplitude of the oscillation in the invariant is comparable to but not necessarily as large as the amplitude necessary to have overlapping of harmonic resonances. APPENDIX

In this appendix, we expand the average Hamiltonian (4.18) about it’s elliptic singularity and use 2nd order perturbation theory to transform to action-angle variables. Repeating (4.18) R = P2 - (P2 - 2PJlf2 Pif2 - i(P2 - 2rj,) Pi” sin $7, ,

we write Hamilton’s

equations

bl = -(aR/adl) i,

= aRIaPI

= g
Defining the elliptic singularity obtain

-

-

2f$l/fw

+

64.2)

P;/z - *(P2 - 2P,) P;1/,

&[P2 - 2PJ P;1’2) sin ti, .

64.3)

p1 , &I as in (3.12) and using (A.2) and (A.3), we 251= Tr/2

$([P,

(A.])

HP2

- Qw)

where (A.5) must be solved numerically pita, and (rj, - 2p1)1/2 about p1 , z& .

G4.4)

= (1 + P2 - 2m9,

b4.5)

for p1 . We expand the functions sin &, ,

MODIFICATION

353

OF INVARIANTS

Defining (‘4.6)

we obtain B = P2 - (P2 + G(p2) q + Z(P,) y

2Q1’2 zy”

- F(P2)

+ F(p2) q

+ A@,) q

+ D(P2) (Ap);cAw)2

+ B(p,)

A’ y””

+ E(P2) q2,

(A-7)

where

G = p;,““(p,

- 2pl)-3’2 + f’y1’“(p2

+ &“‘“(P2

- 2z91’2 - #y’2(P2

- 2P1:)-“2 + zy”“(P2

+ p’;“‘“(P2 + &y’“(P2

E = -

+ &“‘“(-16,

- 24)“2

- 2PJ + P.1’2,

- &P;“‘“(P2

z = gy2(P2

- 2f9-1’2

&P;i2(P2

- 2Pl)4’2

-

+ &““(P2

- 2Q

2&‘,)-5’3

-

-

gy,

pyyp2

-

2pp3

- 2&y

- 2&) f @y”‘“,

- 2Pl).

Since the first three terms on the right in (A.7) do not depend on the variables

354

JAEGER

AND

LICHTENBERG

p 17 Gl, they can be combined with B in determining

the AP - Aw motion with equation for Hamiltonian (A.7) becomes

Pz constant. The Hamilton-Jacobi G(p2) q + Z(P2) q

+ F(p2) q + D(P2)

+ A(p2) (Ap);(Aw)z

q

+ B(p2)

+ E(P2) q

A’

ywy

(A.@

= K(J, , P2).

We can now transform to action-angle variables JIo, 8,O for the linear problem by applying the transformation (3.25) so that Eq. (A.8) becomes

R,(J,O) + 6R,(J1°, 01°) + S2R2(J10, 8,O) = K(J1, a,

(-4.9)

where RO(JIO) = +S2oJ,o, K(JlO, 4O) = h p 0Jl 0 23 (A

R2(~lo,

e,o)

(3083

= x2 1~2~~0q (I R = (F/G)““,

e,o

cos4

e,o

+ $ + $

Go = (FG)1/2,

cos

sin2

e,o

~0~2

e,o

e,o

sin2

, 11 e,o

+ j$- sin4

and to find the additional to obtain

[A(cos?

,

h = (2J1R)“/“/2sZoJ,,

and 6 is an artificial constant measure of smallness used because X,,, the accidental case. Using (3.27), we solve for S, , to obtain S, = - f J,h sin til

e,o

CCG/2 in

&o + 2) + $$ (sin2 elo)],

(A.lO)

term in the energy, we use S, from (A.lO) and (3.29) Kz = 1;2”Jlh2M(p2)

(A.11)

where

The average Hamiltonian

i7 in action-angle

variables J1 , d1valid to X2is

&J, , p2) = & - (6 - 2&)1’2 p:‘2 + F(&) + f-i”J,(l + h2M) (A.12)

MODIFICATION

OF INVARIANTS

355

and the frequency of the energy oscillation, 8, is 8, = aR/aJ, = +@(IQ[l

+ 2P(J, ) Pz, M(PJ].

(A.13)

The nonlinearity in the energy oscillation prevents a secular increasein the action. Finally, substituting (A.1 1) into (3.28) gives the second order term in S(J, , 8,O): S, = JIXz (- f + +

(--60~2

+ 2 sin 3 + g

sin OIocos Or0[I (cos2 flIo + 2)

elo cos

(f

~0~2

elo

+ $ + -$

e 1o[AZ

( -sin2

e,o

- $1

(f cos4 e,o + & cos2elo + $)

elo sin2 elo - + ~0~2

elo

+ i 81

+ $- (A sin4 elo - & sin2e,o - A)] 1,

(A. 14)

from which we obtain the transformation from variables LIP, dw to action-angle variables J1 , 0, correct to second order in 6. As a check of this transformation, it can be substituted into Eq. (A.8) to verify that this equation is satisfied.

ACKNOWLEDGMENT The authors wish to acknowledge the many helpful conversations with Dr. Miles Seidl

REFERENCES 1. R. M. KUL~R~D, Phys. Rev. 106 (1957), 205. 2. G. S. GARDNER, PIzys. Rev. 115 (1959), 791. 3. A. LENARD, Ann. Phys. (N.Y.) 6 (1959), 261. 4. S. CHANDRASEKKAR, “Plasma Physics,” Chap. 3, University of Chicago Press, Chicago, 1960. 5. P. 0. VANDERVOORT, Ann. Phys. 12 (1961), 436. 6. A, J. LICHTENBERG, Plasma Phys. 6 (1964), 135. 7. H. R. LEWIS, JR., J. Math. Phys. 9 (1968), 1976. 8. K. R. SYMON, .I. Math. Phys. 11 (1970), 1320. 9. N. KRYLQV AND N. N. BOGOLIUBOV, “On Nonlinear Mechanics,” Academic Science, USSR, 1945. 10. N. N. BOGOLIUB~V AND Y. A. MIT~OPO~SKY, “Asymptotic Methods in the Theory of Nonlinear Oscillations” (translation), Gordon and Breach, New York, 1961.

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