Differential geometric analysis of radiation-particle interaction
aa r X i v : . [ m a t h - ph ] O c t Differential geometric analysis of radiation-particleinteraction
Kiam Heong Kwa
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USAE-mail: [email protected] , [email protected] Abstract.
On the basis of the Lorentz equations of motion, the orbit of a chargedriven by a generic E.M. field with planar symmetry is formulated and analyzed withinthe framework of a Lorentzian geometry with a curvature whose order of magnitudeis parametrized by the radiation intensity and frequency. This reformulation leads to(i) a demonstration of the integrability of the particle motion in a plane-wave fieldas a result of the vanishing of the curvature, (ii) a manifestation of the parametricdependence of the dynamical response of the particle orbit to the E.M. field on theimpulse factor in terms of local stability and the occurrence of parametric resonanceand (iii) a mathematically precise meaning to the ponderomotive oscillation centerof the charge executing oscillatory motion in a sufficiently low impulsive E.M. field,which is subsequently used to raise (iv) a discussion of the domain of applicability ofthe ponderomotive approximation.PACS numbers: 41.75.Jv, 02.40.-k, 52.20.Dq, 52.35.MwAMS classification scheme numbers: 78A35, 53Z05, 70K20, 70K28, 70K70
1. Introduction
The starting point for many investigations on nonlinear radiation-matter interaction hasbeen the orbit of a single charged particle in an E.M. plane wave studied on the basis ofthe Lorentz equations of motion or its equivalent Lagrangian formulation. For example,see [1, 2, 3, 4, 5, 6]. The primary aims of this paper are (1) to give an exact differentialgeometric formulation of the Lorentz equations of motion in an E.M. field with planarsymmetry and (2) to analyze the dynamics of the orbits of charged particles in responseto such an E.M. radiation within the geometrical framework. It is worth noting that theclass of E.M. fields with planar symmetry includes arbitrary plane-wave fields, standing-wave fields, and E.M. fields of two-counter-propagating waves as particular instances.See (3.1), (4.3), and (5.1) below. It is shown that the complexity of particle dynamicscan be encapsulated economically into a curvature function and associated geometricalquantities.
To demonstrate this conceptual novelty, we show that the integrability ofthe particle motion in an arbitrary E.M. with planar symmetry is a consequence ofthe vanishing of the curvature; the widely known admission of analytic solutions byifferential geometric analysis of radiation-particle interaction the Lorentz equations for arbitrary plane-wave fields is therefore a manifestation of theflatness of a manifold. Another demonstration of the conceptual novelty, which is more important, isthe availability of the Jacobi equations for geodesic flows and the associated curvturefeatures. This way the stability question of particle orbits is translated into the measureof the strength of the geodesic spread.
To illustrate this, we consider the stabilityof particle orbits in a standing wave field by analyzing the associated Jacobi fieldsand indicate the interrelation of the (in)stability of particle orbits and (non)occureneof parametric resonance. Finally, as an important application of the differentialgeometric framework, we employ a mathematical averaging of the Jacobi equationsand the presence of parametric resonance to discuss the domain of applicability of theponderomotive oscillation center dynamics of charged particles.The outline of this paper is as follows. In Sec. 2, the identification of the physicalorbits of a natural mechanical system with the geodesics in the configuration spacewith a suitable semi-Riemannian metric is established. In Sec. 3, the reduction of theinteraction of a charged particle with a generic planar symmetic E.M. field to a naturalmechanical system is made. This sets up the foundation of the differential geometricanalysis of the radiation-particle interaction in this paper.In Sec. 4, the well-known integrability of the particle motion in a plane-wave fieldis shown as a result of the vanishing of a curvature function. In addition, the analyticsolutions of the Lorentz equations for an arbitrary plane-wave field is obtained from thecorresponding geodesic equations.In Sec. 5, the stability question of the particle motion in an E.M. field of standingwave is studied. The effect of the dynamical response of a particle of charge q and mass m placed in an E.M. field of radiation frequency ω and field amplitude E is very oftenexpressed by the magnitude of the dimensionless impulse factor η = q E mω , (1.1)which is usually also called the electron quiver velocity if the charge is an electron orthe intensity parameter [7, 8]. There is no exception when the particle interacts witha generic E.M. field with planar symmetry. However, we will show in this paper that the stability of the orbit of a particle interacting with the E.M. field is not completelydetermined by the order of magnitude of the impulse factor η , but by the differentnumerical subintervals within which the value of η lies. In particular, a highly impulsiveE.M. field ( η ≫
1) may not lead to instability of orbits of particles. This, as we willshow, is a consequence of the (non)occurrence of parametric resonance in the systemconsisting of the particles and the E.M. field, where the impulse factor η is the parameterof the system that determines the (non)occurence of such a parametric resonance.We also employ the Jacobi equations to consider the applicability of the oscillationcenter dynamics of charged particles. The oscillation center dynamics of a chargedparticle in an E.M. field, governed by the so-called ponderomotive force, has beenthe basis of many investigations of the particle motion in response to the field ifferential geometric analysis of radiation-particle interaction η ≪ when a parametric resonanceoccurs, the ponderomotive approximation breaks down. Finally, in Sec. 6, we summarize the results of this paper.
2. Semi-Riemannian Formulation of Natural Mechanical Systems
This is a preparatory section in which we derive the identification of the physical orbits ofa natural mechanical system, a dynamical system that can be described by a Lagrangianof the form L (cid:18) x A , dx A dτ (cid:19) = k BC dx B dτ dx C dτ − Φ( x A ) , (2.1)where k AB is a metric based kinetic energy tensor and Φ is a potential, all of whichdepend only on the coordinates x A , with the geodesics in the configuration spaceendowed with a suitable semi-Riemannian metric. We will also recall the geodesicequations and the Jacobi equations for geodesic spread for later reference. As aconvention, all capital Latin letters A, B, C, and so on, whenever appearing as subscriptsor superscripts, always run over the values 0 , , · · · , N −
1, where N is the number ofdegrees of freedom of the original dynamical system.As is well-known, a natural mechanical system can also be described by theHamiltonian H ( p A , x A ) = k BC p B p C x A ) , (2.2)where p A = ∂ L /∂ ( dx A /dτ ) = k AB dx B /dτ are the generalized momenta, and theHamiltonian is an integral of motion. We are particularly interested in the dynamics oftimelike orbits along which k AB (cid:0) dx A /dτ (cid:1) (cid:0) dx B /dτ (cid:1) = H − Φ < H = H in the case k AB is Lorentzian. In this case, it can be shown that the timelikeorbits are geodesics in the configuration space endowed with the metric tensor g AB = − H − Φ) k AB , (2.3)called the Jacobi metric. This is a classical result for the Riemannian case, a derivationbased on Hamiltonian’s least action principle of which can be found in [15, 16, 17, 18].However, we show that it comes as a consequence of the duality of the Hamiltonian H and the orbit parameter τ . Explicitly, we consider τ as an additional degree of freedomand reparametrizes the orbits by a new parameter λ with the property that dτ /dλ > ifferential geometric analysis of radiation-particle interaction R τ ( λ ) τ ( λ ) L dτ = R λ λ L ( dτ /dλ ) dλ , the Lagrange’s equations in the x A -directions forthe extended Lagrangian L e (cid:18) x A , dx A dλ , dτdλ (cid:19) = L (cid:18) x A , dx A dλ dτdλ (cid:19) dτdλ = k BC dx B dλ dx C dλ dλdτ − Φ( x A ) dτdλ (2.4)are equivalent to the ones for the original Lagrangian L . Then since τ is cyclic withrespect to L e , ∂ L e ∂ ( dτ /dλ ) = − k AB dx A dλ dx B dλ (cid:18) dλdτ (cid:19) − Φ (2.5)is an integral of motion. In fact, comparing (2.2) and (2.5), we see that L e has the value − H , so that k AB dx A dλ dx B dλ (cid:18) dλdτ (cid:19) + Φ = H. (2.6)Solving for dτdλ = s k AB H − Φ) dx A dλ dx B dλ (2.7)from (2.6) and applying the classical Routh’s procedure yields the Routhian L es (cid:18) x A , dx A dλ (cid:19) = L e − ∂ L e ∂ ( dτ /dλ ) dτdλ = − r H − Φ) k AB dx A dλ dx B dλ (2.8)that acts also as a Lagrangian for timelike orbits of the original dynamical system. Thisimplies that the timelike orbits are geodesics of the configuration space endowed with theJacobi metric (2.3). In local coordinates, the geodesics equations are given by d x A ds + Γ ABC dx B ds dx C ds = 0 , (2.9)where s ( τ ) = Z ττ r − g AB dx A dλ dx B dλ dλ (2.10)is the usual affine parameter andΓ ABC = g AD g DB,C + g DC,B − g BC,D ) (2.11)are the Christoffel symbols. As is widely recognized, the local stability of a fiducialgeodesic is governed by the Jacobi separation field J A that evolves according to theJacobi equations ∇ J A ds + R ABCD dx B ds J C dx D ds = 0 , (2.12) ifferential geometric analysis of radiation-particle interaction ∇ J A ds = dJ A ds + Γ ABC dx B ds J C (2.13)is the covariant derivative of J A and R ABCD = Γ
ABD,C − Γ ABC,D + Γ
AEC Γ EBD − Γ AED Γ EBC (2.14)are the components of the Riemann curvature tensor. More explicitly, the Jacobiequations are given by d J A ds + 2Γ ABC dx B ds dJ C ds + Γ ABC,D dx B ds dx C ds J D = 0 . (2.15)This way we have translated the stability problem of orbits in the original mechanicalsystem into geometric language.Specializing to the case k AB = η AB , so that g AB = e σ η AB , where σ = ln [2(Φ − H )]2 (2.16)and [ η AB ] = − · · · · · · · · · · · · · · · (2.17)is the usual Lorentzian metric tensor, we haveΓ ABC = δ AB σ ,C + δ AC σ ,B − η BC σ ,A , (2.18)Γ ABC,D = δ AB σ ,CD + δ AC σ ,BD − η BC σ ,A,D , (2.19)and R ABCD = δ AD ( σ ,BC − σ ,B σ ,C ) − δ AC ( σ ,BD − σ ,B σ ,D ) + ( δ AD η BC − δ AC η BD ) σ ,E,E + η BC ( σ ,A,D − σ ,A σ ,D ) − η BD ( σ ,A,C − σ ,A σ ,C ) . (2.20)So the geodesic equations are given by d x A ds + 2 dσds dx A ds − σ ,A η BC dx B ds dx C ds = 0 (2.21)and, upon using the relation ds/dτ = e σ that follows from (2.6), (2.10), and (2.16), by d x A dτ − σ ,A η BC dx B dτ dx C dτ = 0 (2.22)while parametrized by the original orbit parameter τ . Using (2.6) with λ = τ and k AB = η AB , it can be shown that (2.22) is equivalent to d x A dτ + σ ,A e σ = 0 . (2.23) ifferential geometric analysis of radiation-particle interaction σ ,A e σ = Φ ,A (2.24)by (2.16), (2.23) is in turn equivalent to d x A dτ + η AB Φ ,B = 0 . (2.25)These are simply the Euler’s equations associated with the original Lagrangian L with k AB = η AB .On the other hand, since e σ η AB (cid:0) dx A /ds (cid:1) (cid:0) dx B /ds (cid:1) = − d J A ds + 2 dσds dJ A ds + 2 dx A ds dds (cid:0) σ ,B J B (cid:1) − σ ,A η BC dx B ds dJ C ds + σ ,A,B J B e − σ = 0 . (2.26)Using the relation ds/dτ = e σ , the Jacobi equations are also given by d J A dτ + 2 (cid:18) dx A dτ σ ,B − σ ,A η BC dx C dτ (cid:19) dJ B dτ + (cid:18) dx A dτ σ ,BC dx C dτ + σ ,A,B e σ (cid:19) J B = 0 . (2.27)Denoting the coordinate x by t , we call t the lab time and an orbit along which x A remains constant for all A = 0 a spatially constant orbit . On a spatially constant orbit, σ ,A vanishes identically for all A = 0 by any of the geodesic equations (2.21), (2.22), or(2.23), and dt/dτ = e σ by (2.6) and (2.16). Hence on such an orbit, the Jacobi equationscan be reduced to d J A ds + 2 dσds dJ A ds + e − σ σ ,AB J B = 0 , (2.28) d J A dτ + e σ σ ,AB J B = 0 , (2.29)and d J A dt + dσdt dJ A dt + σ ,AB J B = 0 (2.30)while being parametrized by the affine parameter s , the original orbit parameter τ , andthe lab time t respectively.
3. Dynamics of a Charge in a Generic E.M. Wave with Planar Symmetry
We begin this section with the reduction of the interaction of a charged particle and ageneric E.M. field with planar symmetry to its longitudinal degrees of freedom. Then weshow that the motion of the particle in the direction longitudinal to the field propagationis a natural mechanical system and hence the corresponding orbits are the geodesics inthe reduced configuration space with a suitable Jacobi metric.A generic E.M. with planar symmetry is defined by the vanishing of the longitudinalcomponents of its vector potential and its sole dependence on the longitudinalcoordinates: A t = A x = 0 , A y = A y ( t, x ) , A z = A z ( t, x ) . (3.1) ifferential geometric analysis of radiation-particle interaction m d x α dτ = qη αβ (cid:18) ∂A γ ∂x β − ∂A β ∂x γ (cid:19) dx γ dτ , (3.2)the vector potential A α determines completely the dynamics of a particle of mass m and charge q in the E.M. field. As a convention, all Greek letters α, β, γ , and so on,whenever appearing as subscripts or superscripts, always run over the values 0 , ,
2, and3. We will also use capital and lower-case Latin letters to denote the longitudinal andtransverse degrees of freedom respectively.The solutions of the Lorentz equations are the world lines( x α ( τ )) = ( x A ( τ ); x a ( τ )) = ( t ( τ ) , x ( τ ); y ( τ ) , z ( τ )) , (3.3)each of which is parametrized by the particle’s proper time τ . The coordinate t = x isthe lab time in units of light traveling distance and is related to the conventional time t conv by t = ct conv . (3.4)As is generally known, the Lorentz equations are the Euler-Lagrange’s equations of theLagrangian L (4) (cid:18) x α , dx α dτ (cid:19) = 12 η βγ dx β dτ dx γ dτ + qm A β ( x α ) dx β dτ (3.5)with the equivalent Hamiltonian H (4) ( p α , x α ) = η βγ (cid:0) p β − qm A β (cid:1) (cid:0) p γ − qm A γ (cid:1) , (3.6)where p α = ∂ L (4) /∂ ( dx α /dτ ) = η αβ dx β /dτ + qA α /m are the generalized momenta. Asa consequence of the synchronization of the laboratory clock and a clock comoving withthe particle whenever the particle is at rest ( dx/dτ = dy/dτ = dz/dτ = 0) in the labframe, it follows that H (4) = − .The planar symmetry of the vector potential induces the cyclic nature of thetransverse coordinates y and z with respect to the Lagrangian L (4) . This leads tothe fact that the transverse momenta P a = η ab dx b dτ + qm A a ( t, x ) (3.7)are integrals of motion besides the Hamiltonian H (4) . Thanks to these integrals ofmotion, we can reduce the number of degrees of freedom of the particle dynamics usingthe classical Routh’s procedure. To this end, we solve for dx a dτ = (cid:18) P b − qA b m (cid:19) η ba (3.8)from (3.7) and substitute them into L (2) (cid:18) x A , dx A dτ (cid:19) = L (4) − ∂ L (4) ∂ ( dx A /dτ ) dx a dτ = η BC dx B dτ dx C dτ − Φ( t, x ) , (3.9) ifferential geometric analysis of radiation-particle interaction t, x ) = η ab (cid:0) P a − qm A a (cid:1) (cid:0) P b − qm A b (cid:1) , (3.10)to get a Lagrangian L (2) for the motion of the particle in the direction longitudinal tothe field propagation.The Routh’s reduction indicates that the dynamics of the E.M. field-acceleratedparticle is controlled entirely by the equation d x A dτ = − η AB Φ ,B , (3.11) which are the Euler-Lagrange’s equations for the reduced Lagrangian L (2) . By contrast, subsequent to the launching of the particle, the transverse degrees of freedom have noactive effect on the particle dynamics in the tx -plane and are governed by the longitudinaldynamics through (3.7); that is, the transverse degrees of freedom enter the longitudinaldynamics of the particle only as parameters by means of the initial data.On the other hand, upon comparing (3.9) to (2.1), we note that more significantis the fact that the longitudinal dynamics is a natural mechanical system with theHamiltonian H (2) ( p A , x A ) = η BC p B p C t, x ) , (3.12)where p A = ∂ L (2) /∂ ( dx A /dτ ) = η AB dx B /dτ are the generalized momenta. Also, alongany physical orbit, we have H (2) = H (4) = −
12 (3.13)by (3.6) and (3.10). In consequence, the longitudinal dynamics can be and will beidentified with the geodesics in the tx -plane endowed with the metric tensor (2) g AB = (1 + 2Φ) η AB = e σ η AB , (3.14)as is given in the more general case by (2.3).Being defined on a two-dimensional manifold, the geodesic equations have thesimple form d uds + 2 σ ,u (cid:18) duds (cid:19) = 0 , (3.15 a ) d vds + 2 σ ,v (cid:18) dvds (cid:19) = 0 (3.15 b )while parametrized by the affine parameter s , where u = t − x and v = t + x, (3.16)are the null coordinates. The components of the Riemann curvature tensor can bereduced to (2) R ABCD = e σ K (cid:0) δ AC η BD − δ AD η BC (cid:1) , (3.17) ifferential geometric analysis of radiation-particle interaction K = − e − σ σ ,A,A (3.18)is the Gaussian curvature. Thus the Jacobi equations (2.12) take the form ∇ J A ds + e σ K (cid:0) δ AC η BD − δ AD η BC (cid:1) dx B ds J C dx D ds = 0 . (3.19)This way one sees that the stability question of the orbit of a charged particle in responseto an E.M. field with planar symmetry is encapsulated into the product of two geometricalquantities, i.e., the conformal factor e σ and the Gaussian curvature K .
4. Dynamics of a Charge in a Curvature-Free Wave Field with PlanarSymmetry
It is clear from (3.18) that the tx -plane endowed with the metric (2) g AB is flat if and onlyif σ ( u, v ) = U ( u ) + V ( v ), where U and V are smooth functions of one variable. Underthis circumstance, initially parallel geodesics preserve their separations [19, § K , the geodesic equations (3.15 a )and (3.15 b ) decouple from each other and are equivalent to dds (cid:18) e U ( u ) duds (cid:19) = dds (cid:18) e V ( v ) dvds (cid:19) = 0 . (4.1)Consequently, one has Z e U ( u ) du = const. Z ds and Z e V ( v ) dv = const. Z ds. (4.2)In summary, this shows that the vanishing of the Gaussian curvature K implies theintegrability by quadratures of the particle motion. A curvature-free instance of physical interest is readily furnished by an ellipticallypolarized plane-wave field propagating along the x -axis. The components of the vectorpotential can be represented by A t = A x = 0 , A y ( u ) = E δω cos ωu, A z ( u ) = E √ − δ ω sin ωu, (4.3)where E and ω are the field amplitude and frequency respectively, while δ is apolarization parameter such that δ = 0 , ± δ = ± / √ u = t − x , it is readily seen from (2.16), (3.10), and (3.18) that the induced tx -plane isflat. Thus the widely known integrability of the particle motion is indeed a consequenceof the vanishing of the Gaussian curvature K . Furthermore, the tranverse motion canbe obtained by integrating (3.7), while the longitudinal dynamics of the particle can bedetermined with the aid of (4.2) by integrating Z (cid:20) P y + ηδ cos ωu ) + (cid:16) P z + η √ − δ sin ωu (cid:17) (cid:21) du = const. Z dv, (4.4)where we recall η = q E /mω is the dimensionless impulse factor of the interaction. ifferential geometric analysis of radiation-particle interaction
5. Dynamics of a Charge in a Standing Wave Field with Planar Symmetry
In general, the Gaussian curvature K as determined from (3.18) is nonvanishing. Ofparticular interest is the motion of a charged particle driven by the E.M. field of astanding wave [2, 10, 13] with planar symmetry and polarized linearly along the z -axis.The components of the vector potential are A t = A x = A y = 0 , A z ( t, x ) = E ω sin ωt sin ωx, (5.1)where E and ω are the field amplitude and frequency respectively. In this case, thescalar potential for the longitudinal dynamics is given byΦ( t, x ) = 12 (cid:2) P y + ( P z − η sin ωt sin ωx ) (cid:3) . (5.2)Here we recall that η = q E mω (5.3)is the dimensionless impulse factor of the interaction. The different numericalsubintervals within which the value of η lies, as we will see, play a significant rolein the local stability of the physical orbits of the particle.By (3.11), the longitudinal dynamics is governed by the equations d tdτ = − ηω ( P z − η sin ωt sin ωx ) cos ωt sin ωx, (5.4 a ) d xdτ = ηω ( P z − η sin ωt sin ωx ) sin ωt cos ωx. (5.4 b )If, in addition, we stipulate that P z = 0, then there are two classes of spatially constantorbits. One class of these spatially constant orbits are those along which sin ωx = 0 andwhose x -coordinates are x ( τ ) = ± nπω , n = 0 , , , · · · , (5.5)and another class of spatially constant orbits are those along which cos ωx = 0 andwhose x -coordinates are x ( τ ) = ± (cid:18) n + 12 (cid:19) πω , n = 0 , , , · · · . (5.6)We will use the first class of spatially constant orbits to discuss their local stabilitycharacterized in terms of the numerical subintervals of the impulse factor η . x = ± nπ/ω By setting σ = ln(2Φ + 1) in accordance with (2.16) and (3.13), one finds σ ,t = η ω cos ωt sin ωt sin ωx P y + η sin ωt sin ωx , (5.7 a ) σ ,x = η ω sin ωt cos ωx sin ωx P y + η sin ωt sin ωx , (5.7 b ) ifferential geometric analysis of radiation-particle interaction σ ,tt = ( ηω ) (1 + P y ) cos ωt sin ωx − η sin ωt sin ωx (1 + P y + η sin ωt sin ωx ) , (5.7 c ) σ ,tx = 2( ηω ) (1 + P y ) cos ωt sin ωt cos ωx sin ωx (1 + P y + η sin ωt sin ωx ) , (5.7 d ) σ ,xx = ( ηω ) (1 + P y ) sin ωt cos ωx − η sin ωt sin ωx (1 + P y + η sin ωt sin ωx ) . (5.7 e )In particular, along the spatially constant orbits x = ± nπ/ω , n = 0 , , , · · · , one has σ ,t = σ ,x = 0 and σ ,tt = σ ,tx = 0 ,σ ,xx ( t, x ) = ( ηω sin ωt ) P y . (5.8)It follows from these equations and (2.30) that the Jacobi equations associated with thespatially constant orbits x = ± nπ/ω , n = 0 , , , · · · , are d J t dt = 0 , (5.9 a ) d J x dt + ( ηω sin ωt ) P y J x = 0 (5.9 b )while being parametrized by the lab time t . Due to their simplicity in form, they readilyadmit further analysis to address the dynamical properties of neighboring orbits of thespatially constant orbits x = ± nπ/ω , n = 0 , , , · · · . By a neighboring orbit of aspatially constant orbit we mean one which begins in a sufficiently small neighborhoodcentered around the initial datum for the particular spatially constant orbit in the phasespace of initial data.Without restricting generality, let P y = 0 and let the fiducial orbit be the spatiallyconstant orbit x = 0. On rescaling the lab time using the optical cycle (i.e., 2 π/ω unitsof lab time), (5.9 b ) becomes d J x dT + (2 πη sin 2 πT ) J x = 0 , (5.10)where T = ωt π (5.11)is the dimensionless time in units of optical cycles. Since (2 πη ) R ∞ sin πT dT = ∞ ,(5.9 b ) is oscillatory in the sense that all its solutions have arbitrarily large zeros [20,Theorem 2.4.1]. In other words, the fiducial orbit x = 0 has infinitely many conjugatepoints. Hence we expect that neighboring orbits intersect the fiducial orbit infinitelyoften, in particular, whenever the corresponding solutions of (5.10) are bounded. Thedistribution of these conjugate points may therefore serve as a physically identifiablediagnostic property of the dynamics of the neighboring orbits of the orbit x = 0 asestimated by (5.10) as compared to (5.4 a ) and (5.4 b )More pertinent to the stability question of the spatially constant orbit x = 0 is thefact that (5.10) is a Hill equation, so that the parametric dependence on the impulse ifferential geometric analysis of radiation-particle interaction η of the boundedness of its solutions can be deduced from standard Floquet theoryas follows. Let J x and J x be the solutions of (5.10) such that J x = dJ x dT = 1 and dJ x dT = J x = 0 at T = 0 . (5.12)Following [21], the characteristic function ϕ ( η ) of (5.10) is defined as half the trace ofits fundamental matrix at a period, i.e., ϕ ( η ) = 12 tr " J x J x dJ x /dT dJ x /dT T = . (5.13)Then it is known that all solutions of (5.10) are bounded if and only if | ϕ ( η ) | < ϕ ( η ) depends generally on the value of η ,so does the boundedness of solutions of (5.10) and hence the stability of the spatiallyconstant orbit x = 0. Viewing the impulse factor η as a parameter of the system, wecall any value of η for which x = 0 becomes unstable a resonant value and say thatthere is a parametric resonance under such a circumstance.In fact, | ϕ ( η ) | − , ∞ ). Thus there existinfinitely many disjoint open subintervals ( η + k , η − k +1 ) ⊂ (0 , ∞ ) , k = 0 , , , · · · , orderedaccording to their left endpoints, in which | ϕ ( η ) | < [21, Ch. VII, § § η lying within these intervals, the spatially constant orbit x = 0 is stable. These subintervals are called the stability zones; the complementary closed subintervals,within the interior of each of which | ϕ ( η ) | >
1, are called the instability zones. As anadditional remark, it follows from a classical result [23] that if η η Z / sin πT dT ≤ π , (5.14)then all solutions of (5.10) are bounded. Hence the first stability zone must be of the form(0 , η − ) for some η − ≥ √ π ≈ . η − ≈ . the orbit x = 0 is stable in a sufficiently low impulsive E.M. field ( η ≪ ).5.2. A Mathematically Precise Averaging Principle for the Jacobi Equation In a sufficiently low impulsive E.M. field ( η ≪ x = 0 can be analyzed using the method of averaging.To this end, we note that (5.10) is equivalent to the linear periodic system d J dT = 2 πηA ( T ) J , where J = " J J = " J x (2 πη ) − dJ x /dT and the coefficient matrix A ( T ) = " − sin πT (5.15) ifferential geometric analysis of radiation-particle interaction d J dT = 2 πηA J , where A = " − (5.16)is the average of A over one of its own cycles or over an optical cycle. The solution ofthe averaged system, i.e., (5.16), with the initial condition J (0) = " J J is readilyobtained as J ( T ) = e πηAT J (0)= " cos √ πηT √ √ πηT − √ sin √ πηT cos √ πηT J J , (5.17)showing that the averaged system exhibits simple harmonic motions with the frequency η/ √ " E E = E = J − J = " J − J J − J be the error in estimating J using J with the initial error " E E = E (0) = J (0) − J (0) = " J − J J − J . (5.18)Calculating E ( T ) from (5.15), (5.16), and (5.18) yields E ( T ) − E (0) = 2 πη Z T [ A ( S )( E ( S ) − E (0))+( A ( S ) − A ) J ( S ) + A ( S ) E (0) (cid:3) dS. (5.19)In consequence, one finds k E ( T ) − E (0) k≤ πη Z T (cid:2) k A ( S )( E ( S ) − E (0)) k + (cid:13)(cid:13)(cid:0) A ( S ) − A (cid:1) J ( S ) (cid:13)(cid:13) + k A ( S ) E (0) k (cid:3) dS ≤ πη Z T q ( E ( S ) − E ) sin πS + ( E ( S ) − E ) dS + πη Z T h(cid:12)(cid:12)(cid:12) cos 4 πS (cid:16) J cos √ πηS + J √ √ πηS (cid:17)(cid:12)(cid:12)(cid:12) +2 q E sin πS + E (cid:21) dS ≤ πη Z T k ( E ( S ) − E (0) k dS + πη (cid:16)(cid:12)(cid:12) J (cid:12)(cid:12) + √ (cid:12)(cid:12) J (cid:12)(cid:12) + 2 k E (0) k (cid:17) T. (5.20) ifferential geometric analysis of radiation-particle interaction k E ( T ) − E (0) k ≤ (cid:12)(cid:12) J (cid:12)(cid:12) + √ (cid:12)(cid:12) J (cid:12)(cid:12) + 2 k E (0) k (cid:0) e πηT − (cid:1) . (5.21)This equation implies that if J and J agree initially, then J ( T ) = J ( T ) + O( ε ) (5.22)uniformly for 0 ≤ T ≤ ln(1 + ε ) / πη in the sense that (cid:13)(cid:13) J ( T ) − J ( T ) (cid:13)(cid:13) ≤ | J | + √ | J | ε (5.23)for any ε >
0. Specifically, J x ( T ) = J x ( T ) + O( ε ) , (5.24)where J x ( T ) = J x (0) cos √ πηT + 1 √ πη dJ x dT (cid:12)(cid:12)(cid:12)(cid:12) T =0 sin √ πηT (5.25)is the average value of J x = J as obtained from the averaged system (5.16), i.e., J x = J , and the initial data for J x and J x have been identified. Hence in a sufficientlylow impulsive E.M. field ( η ≪ ), a charged particle that is launched from a neighboringorbit of the spatially constant orbit x = 0 exhibits an almost simple harmonic motionand oscillates about x = 0 with a frequency close to η/ √ oscillations per optical cycle.5.3. The Landau Decomposition of the Jacobi Equation We can reformulate the Jacobi equation, i.e., (5.10), as d J x dT = 2 π η ( − πT ) J x . (5.26)Thus the dynamics of the Jacobi field J x can be viewed as the motion of a particlesubject to a time-independent potential U ( J x ) = ( πηJ x ) (5.27)and a force f ( J x , T ) = 2 π η cos 4 πT · J x (5.28)which varies in time with the frequency 2 oscillations per optical cycle. The frequency of f is considered high in the sense that it is of higher order of magnitude of the frequencyof the motion of the particle in the absence of the force field f , i.e., 2 ≫ η/ √ η ≪ √
2. It is a common practice in plasma physics to decompose the original motion, ifferential geometric analysis of radiation-particle interaction J x in our case, into a secular component X describing the oscillationcenter orbit and an oscillatory component ξ that is nearly periodic [7, 26, 12, 14]: J x ( T ) = X ( T ) + ξ ( T )with d XdT = − π η ( X + 2 sin πT · ξ ) ,d ξdT = 2 π η cos 4 πT · X. (5.29)We call the decomposition ( X, ξ ) a
Landau decomposition . This is a non-uniquerepresentation of J x which depends on a choice of X and ξ satisfying the last twocoupled equtions of (5.29).Of particular interest is the following exact representation. It is obtained byapplying the initial conditions J x (0) and dJ x /dT | T =0 to X and dX/dT and havingthe oscillatory component ξ and its velocity dξ/dT vanish initially. This gives rise tothe compatibility condition X (0) = J x (0) , dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 = dJ x dT (cid:12)(cid:12)(cid:12)(cid:12) T =0 and ξ (0) = 0 , dξdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 = 0 . (5.30)They determine an exact solution to the last two coupled equations of (5.29) and thusconstitute the exact representation of the Landau decomposition ( X, ξ ) expressed by(5.29).
With the compatibility condition, each Jacobi field J x is uniquely determined bya Landau decomposition and vice verse.5.4. The Averaging of the Landau Decomposition On the assumption that the E.M. field is of low impulse, the Laudau decomposition(5.29) can be analyzed using the method of averaging as in the case of (5.10). Togetherwith the compatibility condition (5.30), we will show that the Jacobi field J x andthe oscillation center X agree on average in a mathematically precise sense, therebyquantifying the smallness of the amplitude of oscillatory component ξ as comparedto that of X . Moreover, the result of the calculation is a preliminary in yielding amathematically solid meaning to the ponderomotive approximation of the oscillationcenter X in a sufficiently low impulsive E.M. field ( η ≪ ifferential geometric analysis of radiation-particle interaction d K dT = √ πηB ( T ) K , where K = K K K K = X (cid:0) √ πη (cid:1) − dX/dTξ (cid:0) √ πη (cid:1) − dξ/dT and the coefficient matrix B ( T ) = − − πT
00 0 0 1cos 4 πT (5.31)has period half an optical cycle. The averaged system corresponding to (5.31) is [24] d K dT = √ πηB K , where B = − − (5.32)is the average of B over one optical cycle. The solution of the averaged system with theinitial condition K (0) = K K K K is given by K ( T )= e √ πηBT K (0)= cos √ πηT sin √ πηT cos √ πηT − √ πηT − √ πηT − sin √ πηT cos √ πηT − sin √ πηT cos √ πηT −
10 0 1 √ πηT × K K K K . (5.33)Note that K ( T ) = K + √ πηK T . In order to be consistent with the stipulationthat the average of ξ vanishes over each optical cycle, we necessitate K = K = 0 . (5.34) ifferential geometric analysis of radiation-particle interaction K ( T ) = K ( T ) = 0 (5.35)identically. This renders K purely oscillatory and thus gives a heuristic justification forthe compatibility condition (5.30). With this condition on any solution of the averagedsystem (5.32), we analyze how good the averaged system approximates the originalsystem (5.31) by considering the error F F F F = F = K − K = K − K K − K K K with the initial error F F F F = F (0) = K (0) − K (0) = K − K K − K , (5.36)where we have set K (0) = K (0) = 0 in view of the compatibility condition (5.30). Asfor (5.19), one finds F ( T ) − F (0) = √ πη Z T [ B ( S )( F ( S ) − F (0))+( B ( S ) − B ) K ( S ) + A ( S ) F (0) (cid:3) dS. (5.37)In view of this, one has k F ( T ) − F (0) k≤ √ πη Z T (cid:2) k B ( S )( F ( S ) − F (0)) k + (cid:13)(cid:13)(cid:0) B ( S ) − B (cid:1) K ( S ) (cid:13)(cid:13) + k B ( S ) F (0) k (cid:3) dS ≤ √ πη Z T (cid:2) ( F ( S ) − F ) (1 + cos πS ) + ( F ( S ) − F ) + F ( S ) sin πS + F ( S ) + 2( F ( S ) − F ) F ( S ) sin πS (cid:3) / dS + √ πη Z T n(cid:12)(cid:12)(cid:12) cos 4 πS (cid:16) K cos √ πηS + K sin √ πηS (cid:17)(cid:12)(cid:12)(cid:12) + (cid:2) F (1 + cos πS ) + F (cid:3) / o dS ≤ √ πη Z T k F ( S ) − F (0) k dS + √ πη (cid:0)(cid:12)(cid:12) K (cid:12)(cid:12) + (cid:12)(cid:12) K (cid:12)(cid:12) + k F (0) k (cid:1) T. (5.38)Hence an application of Gronwall’s lemma [25, Lemma 1.3.3] yields k F ( T ) − F (0) k ≤ (cid:12)(cid:12) K (cid:12)(cid:12) + (cid:12)(cid:12) K (cid:12)(cid:12) + k F (0) k (cid:16) e √ πηT − (cid:17) . (5.39)This equation indicates that if K and K agree initially, then K ( T ) = K ( T ) + O( ε ) (5.40) ifferential geometric analysis of radiation-particle interaction ≤ T ≤ ln(1 + ε ) / √ πη in the sense that (cid:13)(cid:13) K ( T ) − K ( T ) (cid:13)(cid:13) ≤ | K | + | K | ε (5.41)for any ε >
0. In particular, one concludes that X ( T ) = X ( T ) + O( ε ) . (5.42)Here X ( T ) = X (0) cos √ πηT + 1 √ πη dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 sin √ πηT (5.43)is the average value of X = K as obtained from the averaged system (5.32), i.e., X = K , and the initial data for the Landau decomposition ( X, ξ ) and its averagedversion (
X, ξ ) have been identified. Thus, combined with the compatibility condition(5.30), (5.25) and (5.43) yield the salient consequence that the original Jacobi field J x and the oscillation center X in the Landau decomposition ( X, ξ ) agree in the sense ofaverage: J x ( T ) = X ( T ) . (5.44)Hence J x ( T ) = X ( T ) + O( ε ) , (5.45 a ) ξ ( T ) = O( ε ) (5.45 b )uniformly for 0 ≤ T ≤ ln(1 + ε ) / √ πη in the sense that | J x ( T ) − X ( T ) | = | ξ ( T ) | ≤ (cid:18) | J x (0) | + 1 √ πη (cid:12)(cid:12)(cid:12)(cid:12) dJ x dT (cid:12)(cid:12)(cid:12)(cid:12) T =0 (cid:19) ε (5.46)for any ε >
0. This quantifies the smallness of the rapid oscillation ξ as compared tothe oscillation center X . Based on the compatibility condition (5.30), we will quantify the closeness of theoscillation center X in the Landau decomposition ( X, ξ ) and its ponderomotiveapproximation X p , thereby providing a mathematically solid formulation for theponderomotive oscillation center X p . To this end, we first calculate the ponderomotiveoscillation center X p following [7, 26, 12, 14]. More sophisticated derivations that leadto equivalent results can be found in [2, 6]. In addition, the result of this calculationleads to an understanding of the breakdown of the ponderomotive approximation whena parametric resonance occurs. This will be expounded in the next section.While regarding X as constant over a given optical cycle and taking into accountof the zero average of ξ over each optical cycle, we integrate the last equation of (5.29)to obtain ξ p ( T ) = − η πT · X ( T ) . (5.47) ifferential geometric analysis of radiation-particle interaction p ’ to distinguish ξ p obtained this way from the exactrapid small oscillation ξ in the Landau decomposition ( X, ξ ). Then the evolutionequation of X p is obtained by first replacing X and ξ in the second equation of (5.29)by X p and ξ p respectively and then by averaging the resulting equation. The result is d X p dT = − π η X p − π η (cid:18) − η πT (cid:19) sin πT · X p = − π η X p − π η X p = − π η (16 + η )8 X p . (5.48)Consequently, X p ( T ) = X (0) cos r η πηT + r
816 + η πη dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 sin r η πηT. (5.49)Here the initial data for X and X p have been identified. Then, in view of (5.43) and(5.49), one has | X ( T ) − X p ( T ) | ≤ r η | X (0) | + 1 πη (cid:12)(cid:12)(cid:12)(cid:12) dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 ! πηT (5.50)by the mean value theorem. This implies that | X ( T ) − X p ( T ) | ≤ r η | X (0) | + 1 πη (cid:12)(cid:12)(cid:12)(cid:12) dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 ! ε (5.51)and thus X ( T ) = X p ( T ) + O( ε ) (5.52)uniformly for 0 ≤ T ≤ ε/ πη for any ε >
0. Finally, combining (5.41) and (5.51) yields | X ( T ) − X p ( T ) | ≤ "
12 + r η ! | X (0) | + (cid:18) √ (cid:19) πη (cid:12)(cid:12)(cid:12)(cid:12) dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 ε, (5.53)showing that X ( T ) = X p ( T ) + O( ε ) (5.54)uniformly for 0 ≤ T ≤ min (cid:8) ln(1 + ε ) / √ πη, ε/ πη (cid:9) for any ε >
0. This gives amathematically rigorous statement about the closeness of the oscillation center X in theLandau decomposition ( X, ξ ) to its ponderomotive approximation X p , whose evolutionis governed by (5.48) instead, in a sufficiently low impulsive E.M. field ( η ≪ We say that the mathematical averaging principle and the ponderomotive approximationfor the oscillation center X in the Landau decomposition ( X, ξ ) break down when(5.42) and (5.54) respectively become inapplicable as T → ∞ . We will exhibit such ifferential geometric analysis of radiation-particle interaction
20a breakdown when a parametric resonance occurs in the original Jacobi field J x . Thiswe do by first relating the exponential order of J x to the ones of X and ξ . More explicitly,we show that if any of the members of the Landau decomposition ( X, ξ ) has a positiveexponential order ‡ , then so has the other element the same exponential order. Thus itmakes sense to talk about the exponential order of the Landau decomposition ( X, ξ )instead of the exponential order of any of its elements. Furthermore, the associatedJacobi field J x possesses the same exponential order as the Landau decomposition( X, ξ ). On the other hand, when a parametric resonance occurs, the Jacobi field J x hasexponential order µ for almost all initial data, where µ is the positive Floquet exponentassociated with the Jacobi equation, i.e., (5.10). Together with the relation amongthe exponential orders of the Jacobi field J x and its associated Landau decomposition( X, ξ ), this implies that both the oscillation center X and the oscillatory component ξ ,assumed small in the ponderomotive approximation, diverge exponentially in amplitudeas T → ∞ with exponent no smaller than the Floquet exponent µ for almost all initialdata in the case when a parametric resonance in J x takes place. In fact, both dX/dT and dξ/dT also diverge exponentially in amplitude as T → ∞ with no smaller exponentunder the same circumstances. These divergences suggest an inconsistency with (5.42)and (5.54) as T → ∞ , where the oscillation center X in the Landau decomposition ( X, ξ )is bounded in both its averaged and ponderomotive approximations. In other words,these divergences suggest the breakdown of the mathematical averaging principle andthe ponderomotive approximation for X . The primary aim of this section is to providea rigorous mathematical justification of these suspicions.To begin with, let us relate the exponential orders of the Jacobi field J x and themembers in the associated Laudau decomposition ( X, ξ ). To this end, note that theevolution equations, i.e., the last two equations of (5.29), of the Landau decomposition(
X, ξ ) satisfy the following equivalent integral equations X ( T ) = X (0) cos √ πηT + 1 √ πη dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 sin √ πηT − √ πη Z T sin √ πη ( T − S ) sin πS · ξ ( S ) dS,ξ ( T ) = ξ (0) + dξdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 T + 2 π η Z T ( T − S ) cos 4 πS · X ( S ) dS. (5.55)First of all, note that these equations imply that if X ( T ) has exponential order α > for T ≥ , then so have ξ ( T ) and J x ( T ) . To see this, let M ≥ ‡ If for some α, M ∈ R , | f ( T ) | ≤ M e αT for all T ≥
0, then the function f is said to have the exponentialorder α . ifferential geometric analysis of radiation-particle interaction | X ( T ) | ≤ M e αT for all T ≥
0. Then | ξ ( T ) | ≤ | ξ (0) | + (cid:12)(cid:12)(cid:12)(cid:12) dξdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 T + 2 π η M Z T ( T − S ) e αS dS = | ξ (0) | + (cid:12)(cid:12)(cid:12)(cid:12) dξdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 T + 2 π η M α ( e αT − αT − ≤ | ξ (0) | + 2 π η M α + (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) dξdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 + 2 π η M α (cid:19) T + 2 π η M α e αT ≤ (cid:20) | ξ (0) | + 1 α (cid:12)(cid:12)(cid:12)(cid:12) dξdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 + 6 π η M α (cid:21) e αT (5.56)by the second equation of (5.55), from which it follows that | J x ( T ) | ≤ (cid:20) M + | ξ (0) | + 1 α (cid:12)(cid:12)(cid:12)(cid:12) dξdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 + 6 π η M α (cid:21) e αT (5.57)for all T ≥ if ξ ( T ) has exponential order β > for T ≥ , then so have X ( T ) and J x ( T ) . Explicitly, let | ξ ( T ) | ≤ M e βT for T ≥
0, where M ≥
0. Then | X ( T ) | ≤ | X (0) | + 1 √ πη (cid:12)(cid:12)(cid:12)(cid:12) dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 + 2 √ πη Z T e βS dS = | X (0) | + 1 √ πη (cid:12)(cid:12)(cid:12)(cid:12) dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 + 2 √ πηβ ( e βT − ≤ | X (0) | + 1 √ πη (cid:12)(cid:12)(cid:12)(cid:12) dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 + 2 √ πηβ ( e βT + 1) ≤ " | X (0) | + 1 √ πη (cid:12)(cid:12)(cid:12)(cid:12) dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 + 4 √ πηβ e βT (5.58)and | J x ( T ) | ≤ " M + | X (0) | + 1 √ πη (cid:12)(cid:12)(cid:12)(cid:12) dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 + 4 √ πηβ e βT (5.59)for all T ≥ when a parametric resonance occurs in the original Jacobi field J x , i.e., when thecharacteristic function ϕ ( η ) of (5.10) has an absolute value | ϕ ( η ) | > both X and ξ diverge exponentially in amplitude as T → ∞ with exponent no smaller than µ foralmost all initial data for J x , where µ is the positive Floquent exponent associatedwith the Jacobi equation (5.10). As we have pointed out earlier, this suggests thebreakdown of the mathematical averaging principle and the ponderomotive approximationfor the oscillation center X because there both the averaged and the ponderomotiveapproximations of X remain bounded for all time.ifferential geometric analysis of radiation-particle interaction X and ξ in the case of parametricresonance, recall from standard Floquet theory that there is a fundamental set of Jacobifields that take the Bloch-Floquet forms J ( T ) = e µT K ( T ) and J ( T ) = e − µT K ( T ),where K and K are periodic functions with the optical cycle being their commonperiod [22]. Let J x = c J + c J , where c and c are integration constants uniquelydetermined by the initial data for J x and vice verse. Thus J x diverges exponentially atthe rate of the Floquet exponent µ for almost all initial data. Let 0 ≤ T < K ( T ) = 0. Then for any integer N > J x ( T + N ) = c K ( T ) e µ ( T + N ) + c K ( T ) e − µ ( T + N ) . (5.60)Consequently, | c K ( T ) | e µ ( T + N ) ≤ | J x ( T + N ) | + | c K ( T ) | e − µ ( T + N ) . (5.61)Hence if X ( T ) has exponential order α > T ≥
0, then it follows from (5.57) and(5.61) that | c K ( T ) | e µ ( T + N ) ≤ M e α ( T + N ) + | c K ( T ) | e − µ ( T + N ) or equivalently | c K ( T ) | ≤ M e ( α − µ )( T + N ) + | c K ( T ) | e − µ ( T + N ) for some M ≥
0. If α < µ , this clearly leads to a contradiction as we pass to thelimit N → ∞ unless c = 0. Thus the oscillation center X must diverge exponentiallyfaster than its associated Jacobi field J x . Indeed, the oscillatory component ξ mustalso diverge exponentially at a rate larger than than of J x . To see this, suppose ξ hasexponential order β > T ≥
0, then it follows from (5.59) and (5.61) that | c K ( T ) | e µ ( T + N ) ≤ M e β ( T + N ) + | c K ( T ) | e − µ ( T + N ) or equivalently | c K ( T ) | ≤ M e ( β − µ )( T + N ) + | c K ( T ) | e − µ ( T + N ) for some M ≥
0. If β < µ , we also reach a contradiction by sending N → ∞ unless c = 0.To conclude this section, we note that the point of departure to validate theexponential divergences of dX/dT and dξ/dT can be taken to be the equations dXdT = −√ πηX (0) sin √ πηT + dXdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 cos √ πηT − π η Z T cos √ πη ( T − S ) sin πS · ξ ( S ) dS,dξdT = dξdT (cid:12)(cid:12)(cid:12)(cid:12) T =0 + 2 π η Z T cos 4 πS · X ( S ) dS (5.62)obtained by differentiating (5.55). Then precisely the same line of argument used toshow the exponential divergences of X and ξ from (5.55) leads one from (5.62) to theconclusion that both dX/dT and dξ/dT diverge exponentially faster than dJ x /dT foralmost all initial data for J x in the case of a parametric resonance. ifferential geometric analysis of radiation-particle interaction (a) η = 0 . ϕ ( η ) ≈ . η = 0 . ϕ ( η ) ≈ . η = 0 . ϕ ( η ) ≈ . η = 0 . ϕ ( η ) ≈ . η = 1 . ϕ ( η ) ≈ − . η = 1 . ϕ ( η ) ≈ − . Figure 1.
The evolution of the Jacobi field J x and its associated Landaudecomposition ( X, ξ ) as well as the oscillation center X and its averaged andponderomotive approximations X = X a and X p respectively for 10 optical cyclesfor different values of η . The initial data are J x = 1 and dJ x /dT = 0. When η ≪ J x ≈ X ≈ X a ≈ X p for many cycles. When η = 1 .
2, the Jacobi field J x experiencesa parametric resonance. In this case, both X and ξ diverge exponentially faster than J x . In consequence, neither X a nor X p serves as a reasonble approximation to X . ifferential geometric analysis of radiation-particle interaction
6. Conclusions
In this paper,(i) we have reformulated and analyzed the dynamical properties of the orbits of chargedparticles interacting with a generic E.M. field with planar symmetry, as governed bythe Lorentz equations of motion, within a differential geometric framework whosegeodesics suffer a mutual deviation in accordance with a curvature induced by thefield intensity;(ii) we have demonstrated the integrability of the particle motion in a plane-wave fieldas a consequence of the vanishing of the curvature;(iii) we have indicated the methodology of examining the local stability of the orbit ofa particle through the Jacobi field within such a geometrical formulation; and(iv) we have also showed the relevance of the geometrical formulation in discussingthe domain of applicability of the ponderomotive oscillation center of a particleexecuting oscillatory motion in the E.M. field.In addition, by considering the motion of a charged particle in a linearly polarizedstanding wave field, we have shown that(a) different numerical subintervals of the impulse factor η give rise to stable or unstableorbits as a consequence of the absence or the presence of parametric resonance ofthe Jacobi field;(b) in a sufficiently low impulsive E.M. field ( η ≪ Acknowledgments
The author wishes to acknowledge fruitful and interesting discussions with ProfessorUlrich H. Gerlach.
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