The Ponderomotive Lorentz Force

The ponderomotive Lorentz force

Graham Cox∗ and Mark Levi†

February 25, 2020

Abstract This paper describes a curious phenomenon: a particle in a rapidly varying potential is subject to an eﬀective magnetic-like force. This force is in addition to the well-known ponderomotive force, but it has not been shown to exist before except for the linear case of a rapidly rotating quadratic saddle potential. We show that this is a universal phenomenon: the magnetic-like force arises generically in potential force ﬁelds with rapid periodic time dependence, including but not limited to rotational dependence.

1 Introduction

The main point of the paper It is a fundamental fact of nature that a changing electric field creates a magnetic field—for instance, a rotating electric dipole creates a magnetic field perpendicular to the plane of rotation. Remarkably, an analogous situation arises in mechanics: a changing mechanical force field creates a magnetic-like force upon a particle, provided that the change is of high frequency. An example,

arXiv:1707.04970v2 [math.DS] 24 Feb 2020 Figure 1: The value of a rotating potential evaluated at a co-rotating point is constant in time: −1 U(R(τ)y, τ) = U0(y), and hence U(x, τ) = U0(R (τ)x).

arising in numerous physical settings, is the motion of a particle in a saddle potential undergoing rapid rotations, as shown in Figure 1. A rotating potential U(x, t) is obtained by composing a ﬁxed 2 potential U0(x), x ∈ R , with the rotation through a rapidly changing angle t/ε:

−1 U := U0 ◦ R (t/ε),

∗Department of Mathematics and Statistics, Memorial University of Newfoundland. [email protected] †Department of Mathematics, Pennsylvania State University. [email protected]

1 (see the caption of Figure 1) and the particle moves according to Newton’s second law:

x¨ = −∇U(x, t/ε), (1) where ∇ is the gradient with respect to x. A remarkable phenomenon, known since Brouwer [1], 2 2 is the stabilization of a saddle, such as U0(x) = (x1 − x2)/2, under its rapid rotations (small ε), as illustrated in Figure 2. The eﬀective force pulling the particle towards the origin and responsible for the stabilization is referred to as the ponderomotive force [12].

2 2 Figure 2: Motion in a rotating quadratic saddle potential with U0(x) = (x1 − x2)/2: (a) is a trajectory of the original equation (1) with rapid rotation; (b) is a trajectory of the averaged equation (7) including terms up to O(ε3); and (c) is a trajectory of the averaged equation without the O(ε3) terms. The “wiggles” in (a) are due to the rapid rotation of the potential; the guiding center transformation (6) smooths these out while retaining lower-order properties of the orbit, in particular its precession. Comparing (b) and (c), we see that the magnetic term, while only of order 3, is responsible for this precession. All three plots have ε = 0.35, x(0) = (1, 0),x ˙(0) = (0, 2) and 0 ≤ t ≤ 430. In particular, (c) shows a periodic orbit over the same time interval as (a) and (b), demonstrating the dynamical signiﬁcance of the magnetic term.

This restoring force, however, does not account for the precessional motion shown in Figure 2. It turns out that there is an additional effective force responsible for this precession. This force has the same mathematical expression as the Lorentz magnetic force exerted on a charged particle moving in a magnetic field, with the “magnetic field” here given in terms of the potential. The ponderomotive Lorentz force thus seems to be be a fitting name for this magnetic-like force. The existence of this force in the linear case (more specifically, for the quadratic saddle potential rotating with a large constant angular velocity) was demonstrated in [7]. The main point of this note is to show that this ponderomotive magnetism is in fact a ubiquitous phenomenon: it arises generically in potential force fields with rapid periodic time dependence, including but not limited to rotational dependence. More specifically, in the main theorem we assign to any solution x = x(t) of (1) with an arbitrary time-periodic potential U(x, τ) its “guiding center” X = X(t, ε), which extracts the average motion of x by discarding the high-frequency oscillatory part, as shown in Figure 2. The fundamental observation (and main point) of this paper is that the guiding center X moves as a charged particle in a time-independent potential field subject to an additional Lorentz magnetic-like force; see Theorem 2 for a detailed statement. The corresponding magnetic vector potential is given explicitly as the time average of an expression involving temporal antiderivatives of U. Although the magnetic effect analyzed in this paper is of one order higher in ε than the ponderomotive force, it can make a qualitative difference in the dynamics, as illustrated in Figures 2 and 4.

2 The ponderomotive force has been well studied, but the ponderomotive Lorentz force does not seem to have been explored apart from the linear case of [7] and a formal treatment of the general case in [13].

Some related background Almost a century has passed since Brouwer analyzed the motion of a particle in a rotating saddle potential [1]. Brouwer considered a particle sliding without friction on a rotating saddle surface in the presence of gravity, and analyzed the linearized equations near the equilibrium. One conse- quence of this analysis was the conclusion that the equilibrium is linearly stable if the two principal curvatures are equal and the rotation is sufficiently rapid. The rotating saddle problem arises in numerous settings in physics, a long list of which can be found in [4] and [6]. We mention only one example here: the motion of asteroids near the equilateral Lagrangian points L4 and L5. The same old problem acquired new relevance starting in the 1990s due to physical applications—a rotating saddle potential can be used to trap charged particles, for instance—and it has since been studied more extensively, both analytically and numerically [2, 5]. Stabilization caused by rapid rotation of the saddle is, at least at the first glance, quite counterintuitive, although it admits a simple heuristic explanation. A related phenomenon which shares this counterintuitive defiance of gravity with the rotating saddle is the stabilization of an inverted pendulum through rapid vibration of its pivot; this was discovered more than a century ago (1908) by Stephenson [14] but is often referred to as the Kapitsa effect, having been studied by Kapitsa [3] in the 1950s. In fact, the inverted pendulum was a motivation for the invention of the Paul trap used to confine charged particles, for which W. Paul was awarded the 1989 Nobel Prize in physics [11]. The Kapitsa effect also has unexpected connections to differential geometry [8] and non-holonomic systems [10]. In this paper we extend the above mentioned results by proving a general averaging theorem which accounts for the effects of rapid periodic changes in a potential force field acting on a particle. According to this theorem, the particle is subject to a magnetic-like force in addition to an effective potential, the latter resulting in the ponderomotive force described above.

Physical applications and motivation A surprisingly large class of physical problems is described by (1). These include the dynamics of electron beams in helical quadrupole magnetic ﬁelds (the motion in the direction of the helical axis results in the rotation of the potential in the transversal plane, so that the displacement along the axis plays the role of the time); the dynamics of ions in rotating ion traps; the restricted circular three-body problem in celestial mechanics, describing the motion near the triangular Lagrange libration points L4 and L5; and much more. A more extensive (but by no means comprehensive) list can be found in [6]. Ponderomotive magnetism arises in all of these problems.

2 Ponderomotive magnetism by averaging

The setting and the notations We consider the motion of a particle in a rapidly oscillating potential,

n x¨ = −∇U(x, t/ε), x ∈ R , (2)

3 where the gradient is taken with respect to x, the function U is periodic in the second variable:

U(x, τ) = U(x, τ + 1), and ε is a small parameter. This general form covers the two special cases of rotating and oscillating potentials mentioned in the introduction. Note that (2) can be written as a Hamiltonian system, with the Hamiltonian 1 H(x, p, t/ε) = p2 + U(x, t/ε). (3) 2 To state our result, we introduce the following temporal antiderivatives of U: Z Z Z V (x, τ) = [U(x, τ) − U(x)] dτ, S(x, τ) = V (x, τ) dτ, A(x, τ) = S(x, τ) dτ, (4) where U = U(x) is the time average of U over one temporal period, and where each of the indeﬁnite integrals is chosen to have zero time average:

V = S = A = 0.

Remark 1. If S is interpreted as the position of a particle at time t, then the antiderivative A is called the absement. Physically, the absement can be thought of as follows: If S is the distance by which the gas pedal of a car is depressed, then the absement is proportional to the total amount of fuel provided, and hence the total amount of energy available to the car.1

n Notation. From now on, for the sake of brevity, the x-derivative of a function f : R → R will be denoted by a prime 0 rather than by ∇. Similarly, the second derivative, i.e. the Hessian, will be denoted by two primes, so that

U 0 = ∇U, U 00 = ∇2U, etc.

The main results The main results of this paper are the following two theorems, the ﬁrst of which describes the averaging in the phase space.

Theorem 1. Assume that the potential U in (2) is of class C3 in x and continuous in t. For any 2n ﬁxed ball in the phase space R there exists ε0 > 0 such that for each positive ε < ε0 there is a time-parametrized family of symplectic transformations

τ τ ψ = ψε :(x, p) 7→ (X,P ) deﬁned on the ball and periodic of period one in τ, such that the time-dependent transformation ψt/ε turns the system with the Hamiltonian (3) into the system with the Hamiltonian

1 ε2 K(X, P, t/ε) = P 2 + U + V 0 · V 0 − ε3S00V 0 · P + O(ε4), (5) 2 2 which is time independent to third order.

The second theorem describes the averaging in the conﬁguration space. For the Newtonian formulation (2) we have the following, recalling the notation from (4).

1We thank Steve Mann for helpful comments regarding physical interpretations of the absement.

4 Theorem 2. Assume that U is of class C4 in x and C1 in t. Let x = x(t) be a solution of (2) n 2 which stays in a ﬁxed ball in R for a time interval I. There is an associated function of the form

X = x + ε2S0(x, t/ε) − 2ε3A00(x, t/ε)x ˙ + O(ε4) (6) that satisﬁes the equation

X¨ = −U 0(X) − ε2W 0(X) + ε3B(X)X˙ + O(ε4), (7)

1 0 0 0 T 0 for all t ∈ I, where W = 2 V · V and B is the skew-symmetric matrix given by B = (b ) − b , with b = S00V 0. We refer to X as the guiding center 3 associated to x, and to (7) as the averaged equation. We do not specify the O(ε4) term in (6) for the sake of simplicity; it can easily be obtained from the proof if desired. Theorem 2 is an extension of the main result in [7] to the nonlinear case and allows for arbitrary periodic time dependence (in particular, the potential need not be rotational). An attempt at a straightforward extension of the method used there quickly becomes too cumbersome, and is probably close to impossible without a computer. We achieve a drastic simpliﬁcation of the proof by using the Hamiltonian nature of the system. The proof is given in Section 4.1.

def Figure 3: b = S00V 0 is the vector magnetic potential: in dimension n = 3, Bb = curl b is the magnetic ﬁeld. In dimension n = 2, the scalar curl b is the magnitude of the magnetic ﬁeld normal to the plane.

Before giving the proof of this theorem, we make several comments. 1. The eﬀective Lorentz force. The term ε3B(X)X˙ in (7) can be interpreted as the force produced by a magnetic ﬁeld upon a charge ε3, as in Figure 3. In dimension n = 3 one has B(X)X˙ = Bb × X˙ , where the magnetic vector Bb = (Bb1, Bb2, Bb3) is associated to the matrix B 3 via the standard isomorphism between R and the Lie algebra so(3):   0 −Bb3 Bb2   B =  Bb3 0 −Bb1  . −Bb2 Bb1 0

Thus ε3B(X)X˙ looks exactly like the standard expression for the Lorentz magnetic force, F = qv × B, with q = ε3, v = X˙ and B = −Bb. 2. The magnetic vector potential. In dimension n = 3 the magnetic field is Bb = curl b. In other words, b = S00V 0 is the magnetic vector potential of the magnetic field. In dimension n = 2, the strength of the magnetic field normal to the plane is given by curl b, the scalar curl 2 in R . 2For many potentials, e.g. superquadratic ones such as x4 − y4, solutions may blow up in finite time. 3although the term is commonly used in a different context: that of motion of a charged particle in magnetic field.

5 3. The ponderomotive Lorentz force causes precession. Consider a rotating potential in dimension n = 2. Since the averaged potential W in this case is rotationally symmetric, every trajectory of the averaged system without the magnetic term that passes through the origin simply oscillates along a straight line segment, as in Figure 4(a). Although the magnetic term is not of leading order in the averaged equation, it is responsible for the precession as illustrated in Figure 4(b) and (c); see also Figure 2. The same dominant effect of precession will manifest itself for trajectories passing near the origin. It should be noted that trajectories not passing through the origin for generic (rotationally symmetric) potentials do precess without the magnetic effect; this precession may dominate the effect of ponderomotive magnetism. However, the ponderomotive magnetism does have a major effect (in the sense of Figure 4) for those trajectories that pass through or near the origin, i.e. for the ones with small angular momenta relative to their energy.

Figure 4: Precessional eﬀect of the ponderomotive Lorentz force: (a) is a trajectory of the averaged system without the magnetic term, while (b) and (c) are trajectories through the origin of the averaged equation with magnetic term included, for ε = 0.35 and ε = 0.5, respectively. Here the potential is U0(x, y) = cos x − cos y, the initial conditions are x(0) = y(0) = y˙(0) = 0,x ˙(0) = 0.2, and the time interval is 0 ≤ t ≤ 700.

4. Linearity in the velocity. Although the original system (2) is fully nonlinear, its averaged counterpart (7) is, to leading order, linear in the velocity X˙ . 5. Possible blow-up. Unlike in the case of quadratic potentials, which correspond to linear systems, solutions may in general blow up in ﬁnite time. An interesting open problem is to understand the nature of the set of initial data for which solutions avoid blow-up. In the case of rotating potentials the problem reduces to studying time-independent potentials with added Coriolis force. We do not discuss this problem since it is peripheral to the main point of this note. 6. An adiabatic invariant. As a side remark we note that the rapid time dependence in the original system (2) gives rise to an approximately conserved quantity. Indeed, for the averaged system (7) the energy ˙ 2 X 2 Eav(X, X˙ ) = + U(X) + ε W (X) (8) 2 is nearly conserved in the sense that along solutions conﬁned to a bounded region one has

d 4 Eav(X, X˙ ) = O(ε ), dt as is easily verified. Now to any solution x of (2) we assign the “effective energy” defined by def E(x, x˙) = Eav(X, X˙ ), where x and X are related via (6), so that d E(x, x˙) = O(ε4) dt

6 along solutions of (2) (which are likewise conﬁned to a bounded region). For small ε this implies small change in E over a long time interval. With some natural assumptions this implies O(ε4−α) change of E(x, x˙) over the time interval 0 ≤ t ≤ ε−α for 0 < α < 4. We leave out all the details since this question lies outside the focus of this paper.

7. An additional adiabatic invariant in the rotational case and near-integrability. For a rotating potential in dimension n = 2, namely U(x, t) = U0(R(−t/ε)x), both averaged potentials U and W , as well as the magnetic vector potential b = S00V 0, are rotationally symmetric, i.e. they depend only on |X|. In this case the truncated system has a conserved quantity associated with this symmetry, namely

Z |X| def 3 Iav(X, X˙ ) = X ∧ X˙ + ε rβ(r)dr, (9) 0 where β = curl b. The first term in this sum is the angular momentum of a unit point mass at X; the integral in (9) is the flux of the magnetic field through the disk of radius |X| (up to the factor of 1/2π). As before, to any solution x of the original system (2) we associate the function def I(x, x˙) = Iav(X, X˙ ), d ˙ 4 where X is related to x via (6). With this definition, dt I(X, X) = O(ε ), and the remark at the end of the preceding paragraph applies. Furthermore, the functions Eav and Iav expressed in terms of position and momentum are in involution. Thus the Hamiltonian form of (7) is a small (in a certain sense) perturbation of a completely integrable Hamiltonian system corresponding to (5) with the O(ε4) terms deleted. It would be of interest to see whether one can apply KAM theory to show that the extended phase space of (7), and thus of (2), has invariant tori whose combined measure approaches full measure as ε → 0 (with appropriate assumptions guaranteeing in particular the avoidance of blow-up). This would imply that most (in the Lebesgue sense) initial data lead to solutions bounded for all time, although one would still expect Arnold diffusion for a small (in the Lebesgue sense) set of solutions.

In the next section we discuss some implications and applications of Theorem 2.

3 Applications of the averaging theorem

The examples of this section give particularly simple illustrations of Theorem 2. These are:

1. Rotating separable potentials;

2. Purely oscillatory potentials;

These two classes encompass a broad range of physical systems—such as the rotating saddle, the Paul trap, and the inverted pendulum with oscillating pivot—and yet are simple enough that the magnetic term in the averaged equation can be worked out explicitly.

The rotating separable potential −1 By a rotating separable potential we mean a rotating planar potential U(x, τ) = U0(R (τ)x), as described in the introduction, with the additional requirement that U0 can be written in polar

7 coordinates as U0(x) = f(r)h(θ), where x = (r cos θ, r sin θ). In this case the time-dependent potential becomes U(x, τ) = f(r)h(θ − τ), (10) where h is a 2π-periodic function. Note that for any function of r and θ − τ, say k(r, θ − τ), that is 2π-periodic in the second variable, the time average satisﬁes 1 Z 2π 1 Z 2π k¯(r, θ) = k(r, θ − τ) dτ = k(r, τ) dτ (11) 2π 0 2π 0 for each θ. That is, the time average k¯ depends only on r. This fact greatly simpliﬁes the following computation, since it tells us that certain terms in the averaged equation will vanish, and hence do not need to be calculated explicitly. We now work out the averaged equation, as given by Theorem 2. The required temporal antiderivatives are V (x, τ) = −f(r)v(θ − τ),S(x, τ) = f(r)s(θ − τ), where v and s are 2π-periodic functions satisfying v0 = h, s0 = v, andv ¯ =s ¯ = 0. Letting er = (cos θ, sin θ) and eθ = (− sin θ, cos θ) denote the standard polar unit vectors, we have

0 0 fh V = −f v er − e r θ (with all terms evaluated at r and θ − τ) and hence 1 1 f 2 W = V 0 · V 0 = (f 0)2v2 + h2 . (12) 2 2 r2 We next compute the angular component of the vector ﬁeld b = S00V 0. (According to (11), the radial component of b will be of the form g(r)er for some function g; therefore its curl vanishes and it does not contribute to the magnetic ﬁeld B in (7).) Using

0 0 fv S = f s er + e r θ ∂ ∂ and the identities ∂θ er = eθ , ∂θ eθ = −er, we compute 0 00 ∂ 0 rf − f S er = S = ve + (∗)er ∂r r2 θ where (∗) denotes a function of r and θ − τ, and 00 1 ∂ 0 1 0 fh S e = S = f s + e + (∗)er, θ r ∂θ r r θ from which we ﬁnd 0 00 0 0 00 fh 00 0 rf − f fh 0 fh S V = −f v(S er) − (S e ) = −f v v − f s + e + (∗)er. r θ r2 r2 r θ Finally, using the relation hs = v0s = −s0v = −v2, we obtain the magnetic vector potential 0 0 2 2 00 0 2ff − r(f ) 2 f 2 b = S V = v − h e + (∗)er, (13) r2 r3 θ where now the coeﬃcient of er only depends on r, by (11). We now examine some special cases using (12) and (13). Throughout we assume that h¯ = 0, so that U = 0 and hence there is no zeroth-order term in the averaged equation (7).

8 Remark 2. For the magnetic term to dominate (7) we must have W constant (so that W 0 vanishes). From (12) we see that this is only possible when f(r) = Ar for some constant A. Then (13) implies −1 that the angular component of b is proportional to r eθ , and so curl b = 0. We conclude that there are no potentials of the form (10) for which the magnetic term is leading order. Suppose f(r) = rm for some real number m, so the potential is homogeneous. In this case we ﬁnd 1 W = m2v2 + h2 r2m−2 2 and 2 2 2 2m−3 b = (2m − m )v − h r eθ + (∗)er In particular, the magnitude of W 0 is proportional to r2m−3 and the magnitude of B is proportional to r2m−4. Recalling the form of the averaged equation

X¨ = −U 0(X) − ε2W 0(X) + ε3B(X)X˙ + O(ε4), we conclude that the magnetic term will dominate ε2W 0(X) when X is small relative to ε. In particular, choosing m = 1/2 we ﬁnd that W and b are proportional to 1/r and 1/r2, respectively. That is, W is equal to the potential of a point change, and b is equal to the vector potential of a magnetic monopole with axis normal to the plane; see Figure 5. For such a potential, assuming that h¯ = 0, the averaged equation is

X JX˙ 0 −1 X¨ = ε2α − 2ε3β +O(ε4),J = , (14) |X|3 |X|3 1 0 | {z } | {z } FCoulomb FLorentz with α and β being the coeﬃents in W and b. An example of such a potential, shown in Figure 5, is √ x2 − y2 U(x) = r cos 2θ = , (15) (x2 + y2)3/4 which has α = 17/64 and β = −13/32.

Figure 5: Rotation of the potential (15) (the graph is shown on the left) gives rise to a “repelling charge” and a “magnetic dipole” acting on the guiding center with forces FCoulomb and FLorentz lying in the plane of motion.

Another case of great physical importance, as described in [4], is the rotating quadratic saddle 2 2 2 potential, i.e. U0(x) = (x1 − x2)/2. This is of the general form (10), with f(r) = r and h(θ) = 1 2 cos 2θ. It follows that 1 1 h¯ = 0, h2 = , v2 = 8 32

9 and so r2 r W = , b = − e + (∗)er, 8 8 θ from which we obtain the averaged equation

ε2 ε3 X¨ = − X + JX˙ + O(ε4), (16) 4 4 in agreement with the result in [7, Theorem 1].

The purely oscillatory potential Another special case of physical importance is

x¨ = −a(t/ε)U 0(x), (17) where a is real-valued and periodic. This equation describes, among other things, the motion of charged particles in the Paul trap [11], in which an electric field is generated by oscillating voltages on electrodes, as shown in Figure 6 (left). The position x of a charged particle in the cavity is then governed by (17). (If gravity is included, an additional constant term should be added to the right-hand side of (17).) With an oscillating voltage applied to the ring, the electostatic field in the cavity looks like the right-hand side of (17). This oscillation can stabilize the equilibrium. By contrast, in a static electric field this equilibrium is necessarily unstable since harmonic potentials cannot have minima. grounded cap V = 0

ring V = V0 sin ωt

endcapV =grounded 0 grounded cap

Figure 6: Left: Axial cross-section of a Paul trap: a ring with two caps, one above and one below. Right: inverted pendulum with a vertically vibrating pivot: a point mass on a weightless rod.

The inverted pendulum with oscillating suspension (Figure 6, right) is also governed by (17). −1 In this case a(t) = −L (g + avert(t)), where avert is the vertical acceleration of the pivot and U(x) = − cos x. For (17), the averaged equation (7) becomes

X¨ = −a U 0(x) − ε2 v2 U 00(x)U 0(x) + O(ε4), where v(τ) = R [a(τ)−a¯] dτ is chosen to have v = 0. Note in particular that the O(ε3) terms vanish and so there is no magnetic term in this case. This adds new insight into the behavior of (17), showing that, at least to third order in ε, there is no precession. This question was not addressed in earlier studies, e.g. [9], which did not compute cubic terms. This example also suggests that the O(ε3) magnetic term is caused by rapid rotation of the potential, rather than oscillation.

10 4 Proof of Theorems 1 and 2

4.1 The general pattern Before proceeding with the proof, we describe the recurring inductive step in the normal form reduction. This step starts with the Hamiltonian

k−1 k k+1 Hk(x, p, t/ε) = H0 + εH1 + ··· + ε Hk−1 + ε H˜k + ε H˜k+1 + ··· , (18) where the time dependence has been eliminated in the terms without the tilde: Hj = Hj(x, p) for j ≤ k − 1, while H˜k = H˜k(x, p, t/ε). Wishing to eliminate the time dependence in H˜k by a time-periodic symplectic transformation (x, p) 7→ (x0, p0), we seek the generating function G(x, p, τ), periodic in τ of period 1, for such a transformation:

0 k+1 0 x = x + ε Gp(x, p , t/ε) 0 k+1 0 (19) p = p + ε Gx(x, p , t/ε).

If G is bounded in the C1 norm in (x, p0), then by the implicit function theorem (19) deﬁnes a map ϕ:(x, p) 7→ (x0, p0), depending on the parameter t, and ϕ−1 is given by

0 k+1 0 0 2k+2 x = x − ε Gp(x , p , t/ε) + O(ε ) 0 k+1 0 0 2k+2 (20) p = p + ε Gx(x , p , t/ε) + O(ε ), where the remainder is a function of x0, p0 and τ = t/ε. The C1-boundedness of G will be established in the course of its construction. The transformed Hamiltonian is given by

0 0 k+1 0 Hk+1(x , p , t/ε) = Hk(x, p, t/ε) + ε ∂tG(x, p , t/ε), (21) where x and p are functions of the new independent variables x0 and p0 by virtue of (20), and where

0 −1 ∂tG(x, p , t/ε) = ε ∂τ G(x, p, τ) with τ = t/ε, so that the correction term in (21) is of order εk:

0 0 k 0 Hk+1(x , p , t/ε) = Hk(x, p, t/ε) + ε ∂τ G(x, p , t/ε). (22)

Substituting (18) and (20) into (22), we obtain the transformed Hamiltonian

k−1 k k+1 Hk+1 = H0 + ··· + ε Hk−1 + ε H˜k + ∂τ G + O(ε ), (23) where all the functions are now evaluated at the new variables (x0, p0). To complete the inductive R step, we make H˜k + ∂τ G time independent by setting G = − (H˜k − H˜ k) dτ, with the constant of integration chosen so as to have G = 0. Note that the terms of order ≤ k − 1 are not aﬀected by the transformation. The Poisson k+1 k bracket {H0,G} appears in the ε -term, while ∂τ G appears in the ε -term; this mismatch is due to the rapid time dependence of the Hamiltonian. When implementing this general scheme we will have to keep track of the higher-order terms, as they will be averaged in subsequent steps.

11 4.2 Proof of Theorem 1 We start with the Hamiltonian 1 H(x, p, t/ε) = p2 + U(x, t/ε). (24) 2 To eliminate time dependence in the ε0-term U, consider a generating function G(1)(x, p, τ) of period 1 in τ (the superscript refers to the ﬁrst step in the averaging). Such a function deﬁnes a symplectic map (x, p) 7→ (x1, p1) implicitly via (1) x1 = x + εGp (x, p1, t/ε) (1) p = p1 + εGx (x, p1, t/ε),

∂ (1) and the transformed system is Hamiltonian with the new Hamiltonian H1 = H + ∂t (εG ) = (1) (1) H + Gτ , where Gτ denotes the derivative with respect to the last variable:

(1) H1(x1, p1, t/ε) = H(x, p, t/ε) + Gτ (x, p1, t/ε). Recalling the form of H, we have

1 2 (1) H1 = p + U(x, t/ε) + G + O(ε). 2 1 τ We now choose G(1) so as to cancel the time dependence in U: Z (1) −G (x, p1, τ) = V (x, τ) := [U(x, τ) − U(x)]dτ,

(1) with the constant of integration chosen so that V = 0. Since such G is independent of p1, the transformation simpliﬁes to

x1 = x 0 (25) p = p1 − εV (x, t/ε), and so the new Hamiltonian is 2 1 2 0 ε 0 0 H1 = p + U(x1) − εp1 · V (x1, t/ε) + (V · V )(x1, t/ε). 2 1 2 To average the O(ε) terms, i.e. make them time independent, we repeat the above transformation, setting

2 (2) x2 = x1 + ε G p(x1, p2, t/ε) 2 (2) p1 = p2 + ε Gx (x1, p2, t/ε) for some new generating function G(2) to be determined. Since the new Hamiltonian will have the form (2) 1 2 0 (2) 2 H2 = H1 + εG = p + U(x2) − εp2 · V (x1, t/ε) + εG + O(ε ), τ 2 2 τ (2) 0 R we choose G (x1, p2, t/ε) = p2 · S (x1, t/ε), where S(x1, τ) = V (x1, τ)dτ is chosen to have zero mean. Therefore the transformation is 2 0 x2 = x1 + ε S (x1, t/ε) 2 00 (26) p1 = p2 + ε S (x1, t/ε)p2.

12 To ﬁnd H2, we ﬁrst compute 2 0 4 x1 = x2 − ε S (x2, t/ε) + O(ε ) and 2 2 2 00 4 p1 = p2 + 2ε p2 · S (x2, t/ε)p2 + O(ε ). Therefore

(2) H2 = H1 + εGτ 1 2 2 00 h 2 0 0 i = p + ε p2 · S (x2, t/ε)p2 + U(x2) − ε U (x2) · S (x2, t/ε) 2 2 2 0 0 ε 0 0 4 − εp1 · V (x1, t/ε) + εp2 · V (x1, t/ε) + (V · V )(x1, t/ε) + O(ε ), 2 which simpliﬁes to 1 2 2 00 0 0 1 0 0 3 00 0 4 H2 = p + U(x2) + ε p2 · (S p2) − U · S + V · V − ε (S V ) · p2 + O(ε ), 2 2 2 with all terms on the right-hand side evaluated at x2 and t/ε. Note that the O(ε) terms have averaged to zero. For the next averaging step we set

3 (3) x3 = x2 + ε Gp (x2, p3, t/ε) (27) 3 (3) p2 = p3 + ε Gx (x2, p3, t/ε).

Since 2 (3) 1 2 2 00 0 0 1 0 0 (3) 3 H3 = H2 + ε G = p + U(x3) + ε p2 · (S p2) − U · S + V · V + G + O(ε ), τ 2 3 2 t we choose Z (3) 0 0 00 1 0 0 1 0 0 G (x2, p3, τ) = U (x2) · S (x2, τ) − p3 · S (x2, τ)p3 + V · V (x2) − V · V (x2, τ) dτ 2 2

(3) R with the constant of integration such that G = 0. Letting A(x2, τ) = S(x2, τ)dτ, with A¯ = 0, we have Z (3) 0 0 00 1 0 0 0 0 G (x2, p3, τ) = U (x2) · A (x2, τ) − p3 · A (x2, τ)p3 + V · V (x2) − (V · V )(x2, τ) dτ. 2

Then the transformation (27) takes form

3 00 x3 = x2 − 2ε A (x2, t/ε)p3 3 (3) (28) p2 = p3 + ε Gx (x2, p3, t/ε) and the corresponding Hamiltonian is

2 1 2 ε 0 0 3 h (3) 0 (3) 00 0 i 4 H3 = p + U(x3) + V · V (x3) + ε G · p3 − U (x3) · G − (S V ) · p3 + O(ε ) (29) 2 3 2 x p

13 where we have used

3 (3) 6 x2 = x3 − ε Gp (x3, p3, t/ε) + O(ε ) 3 (3) 6 p2 = p3 + ε Gx (x3, p3, t/ε) + O(ε ), obtained from (27). Finally, to eliminate the time dependence in the bracketed term in (29) we make one more transformation

4 (4) x4 = x3 + ε Gp (x3, p4, t/ε) (30) 4 (4) p3 = p4 + ε Gx (x3, p4, t/ε) yielding the new Hamiltonian 3 (4) H4 = H3 + ε Gτ . As before, one can choose G(4) to replace the bracketed term in (29) by its average; since G(3) = 0, only the last terms in brackets survives, and

2 1 2 ε 0 0 3 00 0 4 H4(x4, p4, t/ε) = p + U(x4) + V · V − ε S V · p4 + O(ε ). (31) 2 4 2 Combining the transformations, we obtain

3 00 2 0 3 00 4 x4 = x2 − 2ε A (x2, τ)p3 = x + ε S (x, τ) − 2ε A (x, τ)x ˙ + O(ε ); (32) we do not provide an explicit form for the coeﬃcient of ε4 since it is rather cumbersome. This completes the proof of Theorem 1 ♦

3 Remark 3. For the rotating quadratic saddle described in Section 3, the O(ε ) term Gx · p3 − 0 00 0 U (x3) · Gp − (S V ) · p3 is time independent, hence does not need to be averaged. In other words, 4 the coordinate transformation does not contain an ε -term, and so x4 = x3. By contrast, in the general nonlinear case it is necessary to average the ε3-term and to compute the corresponding transformation for x4.

4.3 Proof of Theorem 2 Hamilton’s equations associated with the Hamiltonian (5) are

˙ 3 00 0 4 X = P − ε S V · P P + O(ε ) 0 (33) P˙ = −U 0 − ε2W 0 + ε3S00V 0 · P + O(ε4), where (·)P denotes P -derivative of (·). It is tempting to differentiate the first of the above equations and substitute into the result the expression for P˙ from the second. The problem, however, is that the remainder in the first equation in (33) depends on t/ε and differentiation lowers the order of magnitude to ε3. We remedy this problem by making one more transformation of the Hamiltonian (5) via

5 (5) X1 = X + ε Gp (X,P1, t/ε) (34) 5 (5) P = P1 + ε Gx (X,P1, t/ε),

14 choosing G(5) so as to kill the time dependence in the coeﬃcient of ε4 in the remainder of H. Now (33) transforms into

˙ 3 00 0 4 5 X = P − ε S V · P P + ε E(X,P ) + O(ε ) 0 (35) P˙ = −W 0 + ε3S00V 0 · P + ε4F (X,P ) + O(ε5), where we have dropped the subscripts, writing X,P instead of X1,P1 and where E and F are time independent. 00 0 00 0 Observe that S V · P P = S V . Diﬀerentiating the ﬁrst equation in (35) now yields X¨ = P˙ − ε3(S00V 0)0X˙ + O(ε4). (36)

0 T Note that S00V 0 · P = (S00V 0)0 P , where (·)T denotes the transpose. Substitution of the second equation in (33) into (36) leads to the claim (7) and completes the proof of Theorem 2. ♦

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