Resonant cyclotron acceleration of particlesby a time periodic singular flux tube

# Resonant cyclotron acceleration of particles by a time periodic singular flux tube

J. Asch, T. Kalvoda, P. Šťovíček
###### Abstract

We study the dynamics of a classical nonrelativistic charged particle moving on a punctured plane under the influence of a homogeneous magnetic field and driven by a periodically time-dependent singular flux tube through the hole. We observe an effect of resonance of the flux and cyclotron frequencies. The particle is accelerated to arbitrarily high energies even by a flux of small field strength which is not necessarily encircled by the cyclotron orbit; the cyclotron orbits blow up and the particle oscillates between the hole and infinity. We support this observation by an analytic study of an approximation for small amplitudes of the flux which is obtained with the aid of averaging methods. This way we derive asymptotic formulas that are afterwards shown to represent a good description of the accelerated motion even for fluxes which are not necessarily small. More precisely, we argue that the leading asymptotic terms may be regarded as approximate solutions of the original system in the asymptotic domain as the time tends to infinity.

Keywords: electron-cyclotron resonance, singular flux tube, averaging method, leading asymptotic term

AMS subject classification: 70K28, 70K65, 34E10, 34C11, 34D05

Centre de Physique Théorique (CPT-CNRS UMR 6207) Université du Sud, Toulon-Var, BP 20132, F–83957 La Garde Cedex, France (asch@cpt.univ-mrs.fr).
Supported by Grant PHC-Barrande No. 21899QB, Ministère des affaires étrangères et européennes.

Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Kolejni 2, 120 00 Praha, Czech Republic (tomas.kalvoda@fit.cvut.cz).
Supported by Grant No. LC06002 of the Ministry of Education of the Czech Republic and by Grant No. 202/08/H072 of the Czech Science Foundation.

Department of Mathematics, Faculty of Nuclear Science, Czech Technical University in Prague, Trojanova 13, 120 00 Praha, Czech Republic (stovicek@kmlinux.fjfi.cvut.cz).
Supported by Grant No. 201/09/0811 of the Czech Science Foundation.

## 1 Introduction

Consider a classical point particle of mass and charge moving on the punctured plane in the presence of a homogeneous magnetic field of magnitude . Suppose further that a singular magnetic flux line whose strength is oscillating with frequency intersects the plane at the origin. The equations of motion in phase space are generated by the time–dependent Hamiltonian

 H(q,p,t)=12m(p−eA(q,t))2, with\ A(q,t)=(−b2+Φ(t)2π|q|2)q⊥, (1.1)

where , . Here and throughout we denote for . Our aim is to understand the dynamics of this system for large times. Of particular interest is the growth of energy as well as the drift of the guiding center.

Our main result in the present paper is to exhibit and to prove a resonance effect whose origin can qualitatively be understood as follows. The Lorentz force equals

 ebq′(t)⊥+eE(t)whereE(t)=−Φ′(t)2π|q|2q⊥.

In the induction free case when , the particle moves along a circle of fixed center (cyclotron orbit) with the cyclotron frequency and the cyclotron radius depending bijectively on the energy. If is nonzero but small then the energy, the frequency and the guiding center of the orbit become time dependent. For the time derivative of the energy computed in polar coordinates one finds that

 ddtH=∂∂tH=eq′(t)⋅E(t)=−e2πθ′(t)Φ′(t) (1.2)

and so the acceleration rate is given by

 γ=limτ→∞H(τ)τ=−e2πlimτ→∞1τ∫τ0θ′(t)Φ′(t)dt. (1.3)

One may expect that is close to a periodic function of frequency (which is the case when ). If the frequencies and are resonant, i.e. if there exist indices and in the support of the Fourier transforms of and , respectively, such that then one may speculate that is positive.

Our original motivation to study this problem was to understand the dynamics of so called quantum charge pumps [7, 9] already on a classical level and to gain a detailed intuition on the dynamical mechanisms in the simplest case. We suggest, however, that particle acceleration mechanisms might be of actual interest in various other models, for example in interstellar physics [3, 5]. We speculate that the equations of motion in this domain might exhibit similar ingredients as the ones studied here.

We remark, too, that our results could be of interest in accelerator physics. While the betatron principle uses a linearly time dependent flux tube to accelerate particles on cyclotron orbits around the flux [6], the resonance effect we observe in the present paper has the feature that acceleration can be achieved with arbitrarily small field strength. A second aspect is that, in contrast to the case of a linearly increasing flux, cyclotron orbits which do not encircle the flux tube are accelerated as well. In fact, for the linear case it was shown in [2] that outside the flux tube one has the usual drift of the guiding center, without acceleration, along the field lines of the averaged potential.

In the case of resonant frequencies one readily observes an accelerated motion numerically, see Figure 1 for an illustration. A typical resonant trajectory is a helix-like curve whose center goes out at the same rate as the radius grows. At the moment, a complete treatment of the equations of motion is out of reach. Therefore we first apply a resonant averaging method to derive a Hamiltonian which is formally a first order approximation in a small coupling constant of the flux tube. We then analyze its flow and show a certain type of asymptotic behavior at large times with the aid of differential topological methods. Though this approach gives only an existence result we use the analysis to derive the leading asymptotic terms.

Then, in the second stage of our analysis, these asymptotic terms are used to formulate a simplified version of the evolution equations. We decouple the equations by substituting the anticipated leading asymptotic terms into the right-hand sides. The decoupled system admits a rigorous asymptotic analysis whose conclusions turn out to be fully consistent with the formulas derived by the averaging methods. This holds under the assumption that the singular flux tube has a correct sign of the time derivative when the particle passes next to the origin. It should be emphasized that the singular magnetic flux is not assumed to be very small in the analysis of the decoupled system.

The paper is organized as follows. In Section 2 we summarize our main results. Namely, in Subsection 2.1 we introduce the Hamiltonian equations of motion in action-angle coordinates, in Subsection 2.2 we study the averaged dynamics resulting from the Poincaré-von Zeipel elimination method and in Subsection 2.3 we rederive the leading asymptotic terms by introducing a decoupled system. Finally, in Subsection 2.4 we interpret the formulas derived so far in guiding center coordinates. Proofs, derivations and additional details are postponed to Sections 3, 4, 5 and 6, respectively.

## 2 Main results

### 2.1 The Hamiltonian

In view of the rotational symmetry of the problem we prefer to work with polar coordinates, . Denoting by and the momenta conjugate to and , respectively, one has

 pr=p⋅q/|q|, pθ=p⋅q⊥≡q∧p.

Using that the vector potential is proportional to one finds for the Hamiltonian (1.1) in polar coordinates

 H(r,θ,pr,pθ,t)=12m⎛⎝p2r+(1r(pθ−eΦ(t)2π)+eb2r)2⎞⎠. (2.1)

The angular momentum is an integral of motion and thus the analysis of the system effectively reduces to a one-dimensional radial motion with time-dependent coefficients. From now on we set , and so the cyclotron frequency equals . Put

 a(t)=pθ−12πΦ(t). (2.2)

In the radial Hamiltonian one may omit the term not contributing to the equations of motion and thus one arrives at the expression

For definiteness we assume that .

The time-independent Hamiltonian system, with , is explicitly solvable. In particular, one finds the corresponding action-angle coordinates depending on as a parameter. Substituting the given function for one gets a time-dependent transformation of coordinates. This is an essential step in the analysis of the time-dependent Hamiltonian (2.3) since action-angle coordinates are appropriate for employing averaging methods. Postponing the derivation to Section 3, here we give the transformation equations:

 r = 2√b(I+|a|2+√I(I+|a|)sin(φ))1/2, (2.4) pr = √bI(I+|a|)cos(φ)(I+|a|2+√I(I+|a|)sin(φ))1/2, (2.5)

and conversely,

 I = 12b⎛⎝p2r+(|a(t)|r−br2)2⎞⎠, (2.6) tan(φ) = −1brpr(p2r+a(t)2r2−b2r24). (2.7)

Furthermore, expressing the Hamiltonian in action-angle coordinates one obtains

 Hc(φ,I,t)=bI−|a(t)|′arctan(√Icos(φ)√I+|a(t)|+√Isin(φ)), (2.8)

and the corresponding Hamiltonian equations of motion take the form

 φ′ = b−cos(φ)aa′2√I(I+|a|)(2I+|a|+2√I(I+|a|)sin(φ)), (2.9) I′ = −|a|′2(1−|a|2I+|a|+2√I(I+|a|)sin(φ)). (2.10)

Using action-angle coordinates one can give a rough qualitative description of trajectories in the resonant case. From (2.4) we see that

 r2=12(r2++r2−)+12(r2+−r2−)sin(φ)

where

 r±=√2b(√I+|a|±√I)

are extremal points of the trajectory (see Section 3). As formulated more precisely in the sequel, resonant trajectories are characterized by a linear increase of for large times while . Thus, as the angle increases the radius oscillates between and (though , themselves are also time-dependent). Moreover, since is bounded, implies and as . Therefore the trajectory in the -plane periodically returns to the origin and then again escapes far away from it while its extremal distances to the origin converge respectively to zero and infinity. We refer again to Figure 1 for a typical trajectory in the -plane in the case of resonant frequencies.

Concluding this subsection let us make more precise some assumptions. For the sake of definiteness we shall focus on the case when is positive and greater than the amplitude of and so is an everywhere strictly positive function. Let us stress, however, that this restriction is not essential for the resonance effect as the radial Hamiltonian (2.3) depends only on and thus the sign of is irrelevant for the motion in the radial direction. On the other hand, as discussed in Subsection 2.4, the sign of determines whether the orbit encircles the singular magnetic flux or not.

Furthermore, the frequency of the singular flux tube is treated as a parameter of the model and so we write

 Φ(t)=2πϵf(Ωt) (2.11)

where is a -periodic real function possibly obeying additional assumptions. Hence

 a(t)=pθ−ϵf(Ωt)

where the coupling constant is supposed to be positive and, if desired, playing the role of a small parameter.

### 2.2 The dynamics generated by the first order averaged Hamiltonian

In order to study occurrences of resonant behavior we apply the Poincaré-von Zeipel elimination method which takes into account possible resonances, as explained in detail, for instance, in [1]. In Proposition 2.2 we provide a detailed information on the resonance effect for the dynamics generated by the first-order averaged Hamiltonian.

We start from introducing some basic notation. Let be the -dimensional torus. For and we denote the th Fourier coefficient of by the symbol

 F[f(φ)]k=1(2π)d∫Tdf(φ)e−ik⋅φdφ.

We introduce as the set of indices corresponding to nonzero Fourier coefficients of . For and put

 ⟨f(φ)⟩L=∑k∈LF[f]keik⋅φ.

For example, in the formulation of Proposition 2.2 we use the averaged function , with . Assuming that let us note that it can alternatively be expressed, without directly referring to the Fourier series, as

 ⟨f(φ)⟩Zν=1νν−1∑j=0f(φ+2πνj). (2.12)

To verify the formula observe that the RHS of (2.12) again belongs to . To check that its Fourier series coincides with the LHS of (2.12) it suffices to consider exponential functions , with . In that case one finds that the RHS of (2.12) reproduces the function if is a divisor of and vanishes otherwise.

In this subsection we assume that is given by (2.11) where is regarded as a small parameter and the -periodic real function fulfills

 ∞∑k=1k|F[f(φ)]k|<∞. (2.13)

This implies that .

Following a standard approach to time-dependent Hamiltonian systems we pass to the extended phase space by introducing a new phase and its conjugate momentum thus obtaining an equivalent autonomous system on a larger space. To have a unified notation we rename the old variables , as , , respectively, and set , . With this new notation, we write , (changing this way the meaning of the symbols and in the current subsection).

The Hamiltonian on the extended phase space is defined as

 K(φ1,φ2,I1,I2)=ω2I2+Hc(φ1,I1,φ2/ω2)

Recalling (2.8) and the conventions for and one gets

 K(φ,I,ϵ)=ω1I1+ω2I2+ϵF(φ,I,ϵ) (2.14)

where

 F(φ,I,ϵ)=ω2f′(φ2)arctan(√I1cos(φ1)√I1+pθ−ϵf(φ2)+√I1sin(φ1)). (2.15)

Our discussion focuses on the resonant case when

 λ:=ω2ω1=μν, with\ μ,ν∈N\ coprime, (2.16)

and, moreover, fulfills

 (2.17)

Note that (2.17) happens if and only if is not a constant function. Discussion of the nonresonant case is, at least on the level of the averaged dynamics, simple and we avoid it. Let us just remark that, in that case, it is not difficult to see that trajectories as well as the energy for the averaged Hamiltonian are bounded.

The Hamiltonian (2.14) is appropriate for application of the Poincaré-von Zeipel elimination method whose basic scheme is briefly recalled in Subsection 4.1. The idea is to eliminate from , with the aid of a canonical transformation, a subgroup of angle variables which are classified as nonresonant, and thus to arrive at a new reduced Hamiltonian depending only on the so called resonant angle variables and the corresponding actions. Note, however, that a good deal of information about the system is contained in the canonical transformation itself.

To achieve this goal one works with formal power series in the parameter , and the construction is in fact an infinite recurrence. In particular,  , where the leading term remains untouched,

 K0(ψ,J)=K0(ψ,J)=ω1J1+ω2J2. (2.18)

In practice one has to interrupt the recurrence at some order which means replacing the true Hamiltonian system by an approximate averaged system. In our case we shall be content with a truncation at the first order.

In our model, one can apply the substitution , where are such that ( exist since , are coprime). The phase is classified as nonresonant and can be eliminated while the phase is resonant and survives the canonical transformation. The procedure is explained in more detail in Subsections 4.2 and 4.3. Recalling (2.18) combined with (2.16), here we just give the resulting formula for the averaged Hamiltonian truncated at the first order,

 K(1)(ψ,J,ϵ)=ω1ν(νJ1+μJ2)+ϵK1(ψ,J) (2.19)

where

 K1(ψ,J)=−ω12∑n∈Z∖{0}F[f]−nνinμβ(J1)|n|μein(μψ1−νψ2) (2.20)

and

 β(J1)=√J1J1+pθ. (2.21)

One observes that the averaged four-dimensional Hamiltonian system admits a reduction to a two-dimensional subsystem. Indeed, as it should be, depends on the angles only through the combination . It follows that the action is an integral of motion for the Hamiltonian . The reduced system then depends on the coordinates , .

Moreover, by inspection of the series (2.20) one finds that the Hamiltonian for the reduced subsystem can be expressed in terms of a single complex variable and takes the form , with being a holomorphic function. We refer to Subsection 4.3 for the proof of the following abstract theorem.

###### Theorem 2.1.

Let and suppose is a nonconstant holomorphic function on the open unit disk . Let be a smooth function such that for , and . Let be the Hamilton function on defined by

Then, for almost all initial conditions , the corresponding Hamiltonian trajectory fulfills

 limt→+∞χ1(t)=χ1(∞)∈R,~{}limt→+∞J1(t)=+∞, (2.22)

and

 (2.23)

Theorem 2.1 yields the desired information about the asymptotic behavior of the averaged Hamiltonian system in action-angle coordinates . One has to go back to the original action-angle coordinates , however, since in these coordinates the dynamics of the studied system can be directly interpreted. This means to invert the canonical transformation which resulted from the Poincaré-von Zeipel elimination method. Let us remark that the generating function of this canonical transformation is truncated at the first order as well. Doing so one derives the following result whose proof can again be found in Subsection 4.3.

###### Proposition 2.2.

Suppose that conditions (2.16), (2.17) characterizing the resonant case are satisfied. Let be a trajectory for the first-order averaged Hamiltonian (2.19) but expressed in the original angle-action coordinates, as introduced in Subsection 2.1. Then, for almost all initial conditions ,

 limt→+∞(φ1(t)−ω1t)=ϕ(∞)∈R,~{}~{}limt→+∞I1(t)t=C>0, (2.24)

and

 C=−εω22f′ν(−(ϕ(∞)+π2)λ), with fν(φ)=⟨f(φ)⟩Zν.

### 2.3 A decoupled system and the leading asymptotic terms

In the preceding subsection we studied an approximate dynamics derived with the aid of averaging methods. It turned out that a typical trajectory reaches for large times an asymptotic domain characterized by formulas (2.24). Being guided by this experience as well as by numerical experiments that we have carried out we suggest that formulas (2.24) in fact give the correct leading asymptotic terms for the complete system.

To support this suggestion we now derive the leading asymptotic terms and a bound on the order of error terms for the original Hamiltonian equations (2.9), (2.10) while assuming that the system already follows the anticipated asymptotic regime. At the same time, we relax the assumption on smallness of the parameter .

Applying the substitution

 F(t)=2I(t)+|a(t)|,~{}ϕ(t)=φ(t)−bt,

to equations (2.9), (2.10) we obtain the system of differential equations

 F′(t) = a(t)a′(t)F(t)+√F(t)2−a(t)2sin(bt+ϕ(t)), (2.25) ϕ′(t) = −a(t)a′(t)cos(bt+ϕ(t))√F(t)2−a(t)2(F(t)+√F(t)2−a(t)2sin(bt+ϕ(t))), (2.26)

where (see (2.2) and (2.11)). The real function is supposed to be continuously differentiable and -periodic. The only assumptions imposed on in this subsection are that is not too big so that the function has no zeroes and is everywhere of the same sign as . Recall that for definiteness is supposed to be positive. Clearly, the functions and are bounded on .

Equations (2.25) and (2.26) are nonlinear and coupled together. To decouple them we replace on the RHS of (2.25) and on the RHS of (2.26) by the respective leading asymptotic terms (2.24), as learned from the averaging method. This is done under the assumption that the solution has already reached the domain where is sufficiently large and starts to grow. Thus, to formulate a problem with decoupled equations, we replace in (2.25) by the expected limit value , i.e. the simplified equation reads

 F′(t)=a(t)a′(t)F(t)+√F(t)2−a(t)2sin(bt+ϕ). (2.27)

As stated in Proposition 2.3 below, a solution of (2.27) actually grows linearly for large times. Conversely, equation (2.26) is analyzed in Proposition 2.4 under the assumption that grows linearly. In that case is actually shown to approach a constant value as tends to infinity. For derivation of Proposition 2.4 the periodicity of the functions is not important. It suffices to assume that it takes values from a bounded interval separated from zero. For the proofs see Section 5.

###### Proposition 2.3.

Suppose is -periodic and

 Ωb∈N,~{}f′(−Ωb(ϕ+π2))<0.

Then any solution of (2.27) such that is sufficiently large fulfills

 F(t)=ϵΩ∣∣∣f′(−Ωb(ϕ+π2))∣∣∣t+O(log(t)2)~{}~{}as~{% }t→+∞.
###### Proposition 2.4.

Suppose fulfills

 A1≤a(t)≤A2,\ |a′(t)|≤A3,

for some positive constants , , . Furthermore, suppose has the asymptotic behavior

 F(t)=αt+O(log(t)2)as\ t→+∞, (2.28)

with a positive constant . If obeys the differential equation (2.26) on a neighborhood of , then there exists a finite limit and

 ϕ(t)=ϕ(∞)+O(log(t)t)as\ t→+∞. (2.29)

Remarks. (i) Let us point out an essential difference between equations (2.25) and (2.26). Note that for all such that one has

 ∣∣ ∣∣acos(s)F+√F2−a2sin(s)∣∣ ∣∣≤1, (2.30)

and, consequently, it follows from (2.26) that

 |ϕ′(t)|≤|a′(t)|√F(t)2−a(t)2. (2.31)

Thus the RHS of (2.26) is inversely proportional to . The point is that if the denominator in (2.30) becomes very small for those for which then one gets a compensation by the vanishing numerator. Nothing similar can be claimed, however, for equation (2.25).
(ii) The replacement of the phase by a constant in (2.25) was quite crucial for derivation of the result stated in Proposition 2.3. In fact, suppose that is sufficiently large. In the case of the original equation (2.25), too, essential contributions to the increase of are achieved at the moments of time for which . If equals a constant then these moments of time are well defined and the growth of can be estimated. On the contrary, without a sufficiently precise information about one loses any control on the growth of .

Propositions 2.3 and 2.4 can also be interpreted in the following way. Let us pass from the differential equations (2.25), (2.26) to the integral equations

 F(t)−F(0)−∫t0a(s)a′(s)F(s)+√F(s)2−a(s)2sin(bs+ϕ(s))ds=0, ϕ(t)−ϕ(∞)−∫∞ta(s)a′(s)cos(bs+ϕ(s))√F(s)2−a(s)2(F(s)+√F(s)2−a(s)2sin(bs+ϕ(s)))ds=0.

Suppose satisfies . If is sufficiently large then the functions

 F(t)=αt+F(0), ϕ(t)=ϕ(∞), t>0,

can be regarded as an approximate solution of this system of integral equations with errors of order for the first equation and of order for the second one.

One has to admit, however, that this argument still does not represent a complete mathematical proof of the asymptotic behavior of the action-angle variables , . So far we have analyzed either the dynamics generated by an approximate averaged Hamiltonian in Subsection 2.2 or a simplified decoupled system in the current subsection. Moreover, in the latter case it should be emphasized that the simplified equations were derived under the essential assumption that the dynamical system had already reached the regime characterized by an acceleration with an unlimited energy growth (this is reflected by the assumption that is sufficiently large). Nevertheless, on the basis of this analysis as well as on the basis of numerical experiments we formulate the following conjecture.

###### Conjecture 2.5.

If then the regime of acceleration for the original (true) dynamical system is described by the asymptotic behavior

 I(t)=Ct+O(log(t)2), φ(t)=bt+ϕ(∞)+O(log(t)t)  as\ t→+∞, (2.32)

where is a real constant and

 C=−12ϵΩf′(−ξ)>0,with\ % \ ξ=Ωb(ϕ(∞)+π2). (2.33)

### 2.4 Guiding center coordinates

Being given the asymptotic relations (2.32), (2.33) it is desirable to describe the accelerated motion in terms of the original Cartesian coordinates . The description becomes more transparent if the motion is decomposed into a motion of the guiding center and a relative motion of the particle with respect to this center which is characterized by a gyroradius vector and a gyrophase [8]. Since the presented results have a direct physical interpretation, in this subsection we make an exception and give the formulas using now all physical constants (including and ). Additional details and derivations are postponed to Section 6.

Let be the velocity. We write where, by definition,

 X(q,p,t)=q+1ωcv⊥(q,p,t), R(q,p,t)=q−X(q,p,t)=−1ωcv⊥(q,p,t)

are the guiding center field and the gyroradius field, respectively. In what follows, we use the polar decompositions

 q=r(cosθ,sinθ), X=|X|(cosχ,sinχ), R=|R|(cosϑ,sinϑ).

Concerning the geometrical arrangement, one has the relation

 ϑ=φ+χ−π2  (mod 2π) (2.34)

(with being introduced in Subsection 2.1, for the derivation see Section 6).

The quantities , were introduced (under different names) and studied in [2] where one can also find several formulas given below, notably those given in (2.36). Observe that and, according to (2.3) and (2.6),

Using these relations one derives that

 mω2c2|X|2=H−ωcpθ+eωc2πΦ,  mω2c2|R|2=H. (2.36)

Observe from (2.36) that if is an everywhere positive function then and so the center of coordinates always stays in the domain encircled by the spiral-like trajectory. On the contrary, if is an everywhere negative function then the center of coordinates is never encircled by the trajectory.

For a Hamiltonian trajectory put . Suppose again that and hence . Applying Conjecture 2.5 and recalling (2.11) one has

 I(t)=−e4πΦ′(−1ωc(ϕ(∞)+π2))t+O(log(t)2)~{}~{}as% ~{}t→+∞. (2.37)

Using (2.35) and recalling definition (1.3) of the acceleration rate one finds the positive value

 γ=limτ→∞H(τ)τ=−eωc4πΦ′(−1ωc(ϕ(∞)+π2))>0. (2.38)

From (2.36), (2.35) and (2.37) one also deduces the asymptotic behavior of the guiding center and the gyroradius,

 |X(t)|=√2γtmω2c+O(log(t)2√t), |R(t)|=√2γtmω2c+O(log(t)2√t)  as\ t→+∞. (2.39)

Now it is obvious why the resonance has also consequences for the drift. The growing energy comes along with a growing distance of the guiding center from the origin. Thus in -space the particle oscillates around a drifting center which goes out at the same rate as the radius grows.

From Conjecture 2.5, with a bit of additional analysis (see Section 6), one can also derive consequences for the asymptotic behavior of the angle variables. One has, as ,

 χ(t)=Dlog(ωct)+χ(∞)+o(1),  ϑ(t)=ωct+Dlog(ωct)+ϑ(∞)+o(1), (2.40)

where , and are real constants. Relations (2.39) and (2.40) give a complete information about the asymptotic behavior of the trajectory .

Finally, in connection with formula (2.38) let us point out that the studied system behaves for large times almost as a “kicked” system. Note that, for a fixed ,

 ax−√x2−a2cos(t)=+∞∑n=−∞(x−ax+a)|n|/2eint→2πδ2π(t)~{}~{}as~{% }x→+∞,

where stands for the -periodic prolongation of the Dirac function. Since (see (1.2)), and, in view of (2.4),

 θ′=amr2+ωc2=ωca2(2I+a+2√I(I+a)sinφ)+ωc2

one has

 H′(t)≈−e2Φ′(−1ωc(ϕ(∞)+π2))δ2π/ωc(t+1ωc(ϕ(∞)+π2)).

Thus the system gains the main contributions to the energy growth in narrow intervals around the instances of time for which (mod ). According to (2.34), these are exactly those instances of time when (mod ), i.e. when the particle passes next to the singular flux line.

## 3 Derivation of the Hamiltonian in action-angle coordinates

In this section we derive the transformation equations from coordinates , to action-angle coordinates , .

Let us first note that the equations of motion for the radial Hamiltonian (2.3) have a well-defined unique solution for any initial condition and for all times. They are equivalent to the nonlinear second-order differential equation

 r′′+b24r=a(t)2r3. (3.1)
###### Proposition 3.1.

Suppose is a real continuously differentiable function defined on having no zeros. Then for any initial condition , , with , there exists a unique solution of the differential equation (3.1) defined on the whole real line and satisfying this initial condition.

###### Proof.

Suppose is a solution of the Hamiltonian equations (3.1). Put and . Then

 ∣∣∣ddtH(t)∣∣∣=|a(t)a′(t)|r(t)2≤∣∣∣2a′(t)a(t)∣∣∣H(t).

From here one readily concludes that if is defined on a bounded interval , then there exist constants , , , such that for all . From the general theory of ordinary differential equations it follows that any solution of (3.1) can be continued to the whole real line. ∎

As a first step of the derivation of the transformation we introduce the action-angle coordinates for a frozen time. Assume for a moment that is a constant and denote

 V(r)=a22r2+b2r28.

Suppose a fixed energy level is greater than the minimal value . Then the motion is constrained to a bounded interval , and one has

 E−V(r)=b28r2(r2+−r2)(r2−r2−).

The action equals

 I(E)=1π∫r+r−√2(E−V(ρ))dρ=b8(r+−r−)2.

Hence

 r±=2√b(I+|a|2±√I(I+|a|))1/2=√2b(√I+|a|±√I).

Using the generating function,

 S(r,I)=∫rr−√2(E−V(ρ))dρ=b2∫rr−1ρ√(r2+−ρ2)(ρ2−r2−)dρ,

one derives the canonical transformation of variables between and the angle-action variables . One has

 ∂S∂I = b4∫rr−1ρ⎛⎜⎝ ⎷ρ2−r2−r2+−ρ2dr+(I)2dI− ⎷r2+−ρ2ρ2−r2−dr−(I)2dI⎞⎟⎠dρ = π2−arctan⎛⎜ ⎜⎝r2++r2−−2r22√(r2+−r2)(r2−r2−)⎞⎟ ⎟⎠.

For the angle variable one obtains

 sin(φ)=1√I(I+|a|)(br24−I−|a|2).

Furthermore,

 pr=∂S∂r=b2r√(r2+−r2)(r2−r2−)=2r√I(I+|a|)cos(φ).

Relations (2.4), (2.5), (2.6), (2.7) readily follow.

Let us now switch to the time-dependent case. The Hamiltonian transforms according to the rule