Nonuniqueness of nonrunaway solutions of Abraham-Lorentz-Dirac equation in an external laser pulse
NNonuniqueness of nonrunaway solutions ofAbraham–Lorentz–Dirac equation in anexternal laser pulse
A. Carati a M. Stroppi b August 14, 2020
Abstract
In the paper [1] it was shown that, for motions on a line under theaction of a potential barrier, the third-order Abraham–Lorentz–Diracequation presents the phenomenon of nonuniqueness of nonrunawaysolutions. Namely, at least for a sufficiently steep barrier, the physi-cal solutions of the equation are not determined by the “mechanicalstate” of position and velocity, and knowledge of the initial accelera-tion too is required. Due to recent experiments, both in course andplanned, on the interactions between strong laser pulses and ultra rel-ativistic electrons, it becomes interesting to establish whether such anonuniqueness phenomenon extends to the latter case, and for whichranges of the parameters. In the present work we will consider justthe simplest model, i.e., the case of an electromagnetic plane wave,and moreover for the Abraham–Lorentz–Dirac equation dealt with inthe nonrelativistic approximation. The result we found is that thenonuniqueness phenomenon occurs if, at a given frequency of the in-coming wave, the field intensity is sufficiently large. An analytic es-timate of such a threshold is also given. At the moment it is unclearwhether such a phenomenon applies also in the full relativistic case,which is the one of physical interest.
Keyword : Abraham–Lorentz–Dirac equation, non uniqueness, electronscattering a Universit`a di Milano, Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy b Universit`a di Milano, Corso di laurea in Fisica, Via Celoria 12, 20133 Milano, Italy a r X i v : . [ phy s i c s . c l a ss - ph ] A ug Introduction
Abraham in ref. [2] and Lorentz in ref. [3] (for the later Dirac relativisticversion see ref. [4]), proposed the following equation in order to describe themotion of a radiating electron m ¨ x = F ( x , ˙ x ) + 2 e c ... x , (1)where m , e and c are the mass of the electron, its charge and the speedof light respectively, while F ( x , ˙ x ) is the force acting on the electron, forexample the Lorentz force due to an incoming electromagnetic wave. As iswell known, the solution of the equation for generic initial data diverge as e tε for t → + ∞ : so they are physically absurd, since they keep continuing toaccelerating also if the force (i.e., the electromagnetic pulse) vanishes. Thisinclined theorists to discharge such an equation, replacing it by some suitableapproximationt, for which runaway solutions don’t show up. The so calledLandau–Lifschitz approximation (see ref. [5], or the more recent ref. [6]) wasvery recently tested in some experiments of interaction between a beam ofultra relativistic electrons with strong laser pulses. The agreement betweentheoretical prediction and experimental data was not completely satisfactory(see ref. [7, 8]).One might think that the use of the original equation (1) could givebetter agreement. The proposal advanced by Dirac in order to overcomethe problem of the runaway solutions, was to admit that the only physicallysignificant solutions are the ones for which the accelerations ¨ x tends to zeroas t → + ∞ . From the mathematical point of view, one has then to deal nomore with a Cauchy problem, for which existence and uniqueness of solutionsare granted, but with a boundary value problem, in which are given themechanical data of position and velocity at −∞ , and the acceleration at+ ∞ . For boundary problems uniqueness is not granted, i.e., having fixedthe mechanical data before the interaction, there might exist several solutionswhich satisfy the nonrunaway condition ¨ x →
0. Any approximation, as theLandau–Lifschitz one, which instead admit just one solution, will be a poorone in that regime.So, it is of importance to understand whether, and eventually in whatregime, the nonuniqueness of the nonrunaway solutions shows up for theAbraham–Lorentz–Dirac equation. In this paper we investigate such a prob-lem for the nonrelativistic version (1) of the Abraham–Lorentz–Dirac equa-2ion, in the case of an incoming electromagnetic plane wave. We will showthrough numerical computations that there exists a threshold in the intensityof the field, above which nonuniqueness occurs. Some numerical checks arealso performed, to control whether below threshold the Landau–Lifschitz ap-proximation is sound. It appears that also well below the threshold the twoequations lead to very different behaviors, with the electron radiating, ac-cording to the Dirac equation, much less energy than according the Landau–Lifschitz approximation.The paper is organized as follows. In Section 2 we describe the modelstudied, while in Section 3 we give an analytic estimate of the region ofparameters in which nonuniqueness is expected to occur. In Section 4 weillustrate the numerical results, and in Section 5 a comparison with theLandau-Lifschitz approximation is given. The conclusions follow.
We consider the case of the interaction of an electron, described by theAbraham–Lorentz–Dirac equation, with an external electromagnetic linearlypolarized plane wave. We will take the x axis as the direction of the wavepropagation, the y axis as the direction of electric field and finally the z axis asthe direction of the magnetic field. In the Coulomb gauge, the scalar potentialvanishes, while the vector potential A takes the form A = (0 , F ( x − ct ) , F ( x − ct ) an arbitrary function, and c the speed of light. To be definit, wemodel the electromagnetic pulse by choosing F ( ξ ) = A exp( − ξ / σ ) cos( kξ ),although every choice with F vanishing sufficiently fast at infinity would givethe same qualitative results. So, the electromagnetic field takes the form (cid:40) E ( r , t ) = − F (cid:48) ( x − ct ) ˆ e y B ( r , t ) = F (cid:48) ( x − ct ) ˆ e z , where ˆ e y and ˆ e z are unit vectors directed as the y and z axis respectively,while F (cid:48) denotes the derivativeof F with respect to its argument. Denoting by x ( t ) = (cid:0) x ( t ) , y ( t ) , z ( t ) (cid:1) the electron trajectory, the Abraham–Lorentz–Dirac3quation takes the form m ¨ x = ec F (cid:48) ( x − ct ) ˙ y + mε ... xm ¨ y = eF (cid:48) ( x − ct ) − ec ˙ xF (cid:48) ( x − ct ) + mε ... ym ¨ z = mε ... z , where we have denoted the constant 2 e /mc by ε . Notice that the equationfor z decouples, and that the only nonrunaway solutions are z ( t ) = z + v z t ,i.e., uniform motions. From now on, we consider just the first two equations,which, by defining ξ def = x − ct , can be put in the following form ¨ ξ = emc F (cid:48) ( ξ ) ˙ y + ε ... ξ ¨ y = − emc F (cid:48) ( ξ ) ˙ ξ + ε ... y . (2)The phase space corresponding to such an equation is six–dimensional,but the system can be reduced to a four dimensional one exploiting theinvariance by translation along the y –axis. In fact, the second equation givesdd t (cid:18) ˙ y − emc F ( ξ ) − ε ¨ y (cid:19) = 0 , i.e., the Abraham–Lorentz–Dirac equation reduces to ¨ ξ = emc F (cid:48) ( ξ ) ˙ y + ε ... ξ ˙ y = − emc F ( ξ ) + ε ¨ y + C , where C is an integration constant, depending on the initial data. We caninclude the constant C in the potential, thus defining the ”effective potential” F C ( ξ ) = F ( ξ ) + C , and introduce the new variable v def = ˙ y : in such a way, onegets the equation ... ξ = 1 ε (cid:18) ¨ ξ − emc F (cid:48) C ( ξ ) v (cid:19) ˙ v = 1 ε (cid:18) v + emc F C ( ξ ) (cid:19) , (3)i.e., an equation in a four–dimensional phase space.4o discuss the solution of this equation, consider first the “mechanicalcase” ε = 0, i.e., the case in which emission is neglected. So one gets ¨ ξ = emc F (cid:48) C ( ξ ) vv = − emc F C ( ξ ) , which reduces to the one dimensional Newton’s equation¨ ξ = − e m c F (cid:48) C ( ξ ) F C ( ξ ) , with a potential V C ( ξ ) = e F C ( ξ ) / m c . The solutions are readily found. Inparticular, for motions of scattering type, if the initial “kinetic energy” ˙ ξ / V C ( ξ ), the electron will pass the barrier, whileit will be reflected if the initial kinetic energy will be smaller. In addition, itis easily checked that the zero of F C ( ξ ) gives stable equilibrium points, whilemaxima of the modulus | F C ( ξ ) | will give unstable equilibria.Return now to the full Abraham–Lorentz–Dirac equation (3). As recalledin the introduction, we look for for “exceptional” initial data which corre-spond to solutions having an asymptotically vanishing acceleration. In otherterms, given the initial value ξ and ˙ ξ , we want to find whether initial data v and ¨ ξ exist such that the corresponding solutions of (3) are nonrunaway.Since we are considering a scattering problem, this can be implemented in astraightforward way by numerically integrating backward in time the equa-tions of motion. In other terms, one fixes the final data outside the interactionzone and integrates backwards in time: in such a way the Dirac manifold becomes an actractor, and after a small transient the orbit practically will lieon such a manifold. Once the electron did come back into the non interactingzone, one gets the initial data which gives rise to a nonrunaway solution.Numerical evidence suggests that, if the electromagnetic field F ( ξ ) is“strong” enough, then, having fixed the mechanical data ξ and ˙ ξ , there existseveral initial v and ¨ ξ which give rise to nonrunaway different trajectories. .Such nonuniqueness phenomenon will be discussed in the next Section. The Dirac manifold is defined as the subset of phase space spanned by the non runawaysolutions. In geometric terms, the Dirac manifold is folded. The nonuniqueness phenomenon
Following ref. [1], in order to discover whether there exist several nonrun-away solutions corresponding to the same initial mechanical state, we startinvestigating the unstable equilibrium point. By rescaling time by t → εt ,the equations (3) becomes ... ξ = eε mc F (cid:48) C ( ξ ) v + ¨ ξ ˙ v = − emc F C ( ξ ) + v . (4)The equilibrium points of such an equation can be subdivided into twoclasses: • The point(s) v = 0 , ξ = ξ ∗ with F C ( ξ ∗ ) = 0 (and obviously ˙ ξ = ¨ ξ =0). Such points corresponds to the stable equilibrium points of themechanical case, and are not interesting for the scattering states. • Points v = v ∗ , ξ = ξ ∗ with F (cid:48) C ( ξ ∗ ) = 0 and v ∗ = eεmc F C ( ξ ∗ ).We consider only equilibrium points of the second type, more precisely weconsiders points such that ξ ∗ is a maximum for F C ( ξ ). It turns out that, asthe parameters are changed, such points exhibit a bifurcation from a saddleto a saddle-focus, the same which drives the nonuniqueness phenomenon inthe one dimensional case (see ref. [1]). In fact, putting χ = ξ − ξ ∗ and u = v − v ∗ , the equation (4) to the first order becomes (cid:40) ... χ = − k χ + ¨ χ ˙ u = u , (5)were we defined k = e ε m c F (cid:48)(cid:48) C ( ξ ∗ ) F C ( ξ ∗ ) . (6)So the linearized equations decouple: the second one defines a directionwhich is always unstable, while the first one is the same one just studied for The nonrunaway solutions are the ones which fall on the equilibrium point, and thusthey do not describe scattering states. k cr = 2 √ . (7)For k < k cr the equilibrium point is an unstable saddle, with one stabledirection and two unstable ones. Instead, for k > k cr one gets two complexeigenvectors, i.e., one has again a stable one-dimensional manifold (call itΣ s ), but the unstable manifold is indeed an unstable focus: the points spiralout from the origin going to infinity. This is the source of the nonuniquenessbehavior.In fact, one can argue as follows. Return to the nonlinear equation: theunstable manifold is three–dimensional, while the nonrunaway manifold, asrecalled above, is a two-dimensional one, so that generically there will be aone-dimensional intersection γ ( t ), which will be a solution belonging bothto the unstable manifold and to the nonrunaway manifold: γ ( t ) springs outspiraling from the unstable equilibrium point at t = −∞ , and goes to infinitywith a vanishing acceleration for t → + ∞ .Consider now, at t = + ∞ , the nonrunaway solutions near to γ , and prop-agates them back in time: by continuity of solution of (4) with respect to theinitial data, this solutions will follow γ ( t ) near the equilibrium point spiral-ing about the stable one-dimensional manifold Σ s . The backward-time flowturns the stable direction into the only unstable one, so that the orbits willfinally follow the Σ s manifold returning again to infinity. In other terms, theexistence of an intersection between the unstable manifold and the Dirac one,entails that the Dirac manifold will be wrapped around the stable manifoldΣ s . This is the origin of the non uniqueness property.In fact, fix now ξ = const sufficiently distant from the origin, and con-sider the intersection of the two-dimensional Dirac manifold (before scat-tering) with the three-dimensional hyperplane ξ = const : one would get acurve which projects on the plane of the initial “mechanical data” ( ˙ ξ, v ) likea (deformed) spiral. Letting C changing, the different spiral will in gen-eral intersect giving rise to different nonrunaway trajectory for the samemechanical initial data.These geometric considerations are obviously not a proof, but just anindication that the bifurcation of the unstable equilibrium points could drivethe appearance of the nonuniqueness behavior. In the next Section we will Because in general they will have different center. I n i t i a l K i ne t i c E ne r g y Final Kinetic Energy1.131.13251.135 0.20 0.205 I n i t i a l K i ne t i c E ne r g y Final Kinetic Energy
Figure 1: Plot of the initial kinetic energy vs. the final one for field am-plitude A = 1, and vanishing wave vector. The map is not one to one, andthis implies nonuniqueness of the nonrunaway solutions. Indeed, drawing ahorizontal line at energy about 1 .
13, one immediately checks that to a giveninitial energy there correspond different final ones. The inset hints at thecomplex structure of the maxima and minima of such a curve.show, by numerical computations, that this is indeed the case.
The equations of motion (2) were integrated by a third order Runge–Kuttamethod which is easy to implement and sufficiently fast for our purposes.Moreover, we studied two case: either a simple Gaussian incoming wave eεc F ( ξ ) = Ae − ξ σ , (8)8 cr A=0.45 I n i t i a l K i ne t i c E ne r g y Final Kinetic Energy0.1630.058 0.061 0.064 I n i t i a l K i ne t i c E ne r g y Final Kinetic Energy
Figure 2: Plot of the initial kinetic energy vs. the final one for three fieldamplitudes: A = 0 .
31 smaller than A cr (cid:39) .
38, the critical one and A = 0 . A = 0 .
45 there exists a very weak local maximum (see the inset), whichentails the nonuniqueness of the nonrunaway solutions.or the more complex wave form eεc F ( ξ ) = Ae − ξ σ cos kξ , (9)which allows one to investigate the role of the wave-length in the scatteringprocess. In the latter case one can rescale the distances by the wave lengthof the incoming laser pulse. Then, all the constants of the problem areresumed into only two parameters: the field intensity A and the width σ ofthe electromagnetic pulse.In the pure Gaussian case (8), we have taken σ = 1 and studied thebehavior of the nonrunaway solutions as the field intensity A is changed.In particular, we find that ξ = 0 is an unstable equilibrium point, in fact9he only equilibrium point. We compute the value A cr which corresponds,through the formula (6) to the value of k cr . For σ = 1 one finds A cr (cid:39) . σ = 10, and, by rescaling, k = 1. In this case ξ = 0 is again an unstableequilibrium point, even if now there exists an infinite number of them (bothstable and unstable). The point ξ = 0 gives however a lower value for A cr ,which in this case corresponds to A cr (cid:39) . ξ f , y f , ˙ ξ f , ˙ y f , ¨ ξ f , ¨ y f ) to the initial one ( ξ i , y i , ˙ ξ i , ˙ y i , ¨ ξ i , ¨ y i ). Theonly independent final parameters are the final velocities ˙ ξ f and ˙ y f . In fact,as one is dealing with s scattering case, one has to considered ξ f large (i.e.states in which the electron has left the interaction zone with the laser pulse),i.e., an arbitrary (but fixed) value for | ξ f | = R such that the force due to theelectromagnetic field essentially vanishes. In such a case one is forced to fix¨ ξ f = ¨ y f = 0, by the nonrunaway condition. Moreover, due to the invarianceunder translation along the y axes, one can fix arbitrarily y f = 0.Having fixed the final data, one starts integrating backwards up to atime such that the electron, after having interacted with the electromagneticwave, returns into a zone of vanidhing field, for example again at | ξ i | = R .At this moment one collects the initial value ˙ ξ i , ˙ y i , ¨ ξ i , ¨ y i . So defined, the mapfrom the “final” to the “initial” data is one to one. The problem is whetherthe inverse mape, i.e., the physical one which maps the “initial” data to the“final” ones, is one to one, or not. If it is one to one there is uniqueness, i.e., toa mechanical data of position and velocity corresponds just one nonrunawaysolution; if it is one to many, to a single mechanical state, there correspondsdifferent nonrunaway solutions with different asymptotic final states.In order to answer this question, we made the preliminary step of reducingto the case of a scattering normal to the plane wave, i.e., to the case inwhich the component ˙ y i along the y axis of the initial velocity vanishes. So,one has to solve the equation ˙ y i ( ˙ ξ f , ˙ y f ) = 0 (which is easily solved by thebisection method). This gives ˙ y f as a function of ˙ ξ f , which remains the onlyfree parameter. A curious feature of this equation, probably linked to theconservation of the y component of momentum in the mechanical case, isthat ˙ y f = 0 gives a good approximation to the true solution.Now, by a simple inspection of the curve ˙ ξ i as a function of ˙ ξ f one cancheck whether the inverse map is one to one or not. Equivalently one caninspect the curves of the initial kinetic energy vs. final kinetic energy: in10 I n i t i a l K i ne t i c E ne r g y Final Kinetic Energy2.42.4152.43 0.10 0.15 0.20 I n i t i a l K i ne t i c E ne r g y Final Kinetic Energy
Figure 3: Plot of the initial kinetic energy vs. the final one for field am-plitude A = 1, and non vanishing wave vector. Notice the jump. The insetis an enlargement of the curves around the minimum, which clearly exhibitsthe nonuniqueness phenomenon. There are other jumps, not shown in thefigure, at low energy. The jumps imply that, for some initial energies, thereare no scattering solutions: for such energies the electron falls onto a stableequilibrium point.the nonuniqueness case such map would show a non monotone behavior. Infigure 1 this curve is drawn for the pure Gaussian potential (8) with A = 1,the inset showing details about the local maximum. There is evidence of acomplex sequence of nested maxima, as in the case investigated in ref. [1].So the map is not monotone, and thus the inverse map is not one to one.Figure 2 shows what happens when the field strength A is increased frombelow the critical value to above it: the curves of the initial kinetic energyare reported versus the final ones (always for fixed initial value ˙ y i = 0): if A = 0 .
31 the curve is monotone increasing, so that the inverse map is oneto one and there is uniqueness of the nonrunaway solutions. For A = A cr I n i t i a l K i ne t i c E ne r g y Final Kinetic Energy0.1400.1440.1480.030 0.0325 0.035 I n i t i a l K i ne t i c E ne r g y Final Kinetic Energy
Figure 4: Same as figure 3, for A = 0 .
3, below the critical value. Now theinsets show that the map is one to one, so that one has uniqueness.the curve seems to have an inflection point but one can consider the inversemap again as one to one, i.e., uniqueness of non runaway solutions. Instead,a carefully inspection of the case A = 0 .
41, above the critical value, shows aweak local minimum for a final kinetic energy of (cid:39) .
06 (more evident in theinset of the figure), so that uniqueness is lost.Figure 3 and 4 refers to the case of the potential given by (9). In figure 3the initial kinetic energy is plotted versus the final one for A = 1: the insetis an enlargement about the minimum. Again, above the critical value, themap is not one to one, and one has the nonuniqueness phenomenon. Noticethat the map has a jump, i.e. the inverse map is not defined in a certaininterval. It seem reasonable to assume that for such a value of the initialkinetic energy, the incoming particle falls onto one of the stable equilibriumpoints. This indeed happens for some value of the energy, but an analyticalproof is lacking. A more detailed study at low final energies (too low to beappreciable in the figure) shows that there are other jumps.12 E l e xc t r on y po s i t i on Electron ξ position-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6LL solutionADL solution E l e xc t r on y po s i t i on Electron ξ position Figure 5: Comparison between orbits computed using the Landau–Lifschitzapproximation (broken line), and the ones computed using the full Abraham–Lorentz–Dirac equation (full line) with the same initial mechanical data:upper panel refers to a pure Gaussian field with intensity A = 0 .
31 belowthe critical one and initial energy E = 0 . A = 1 above the critical one, for an initial energy E =1 . A = 0 .
3, which is below the critical value. Now, the map appears tobe one to one, and uniqueness recovered. As in the case of A = 1, the inversemap is not defined for some intervals of the initial kinetic energy. Again wethink this is due to the fact that the particle be captured by one of the stableequilibrium points. 13 E ne r g y l o ss Incoming electron energy 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5A=0.05A=0.25A=0.45 A=0.65A=0.85Incoming electron energy
Figure 6: Energy loss in a collision versus the energy of the incoming elec-tron, for several values of the field intensity, ranging from A = 0 .
05, wellbelow the critical intensity, to A = 0 .
85, above it, in semi logarithmic scale.Left panel: loss computed according Landau-Lifshitz approximation; rightpanel: loss computed according Abraham–Lorentz–Dirac equation. Noticetaht the Abraham–Lorentz–Dirac equation predicts a maximum for the en-ergy loss.
The Landau-Lifschitz approximation is obtained from the Abraham–Lorentz–Dirac equation using the following argument. For small ε one has m ¨ x (cid:39) F ( x , ˙ x ) ; (10)14o that one can obtain an approximation of the third derivatives by... x = dd t ¨ x (cid:39) dd t (cid:32) m F ( x , ˙ x ) (cid:33) , which substituted into the Abraham–Lorentz–Dirac equation gives, neglect-ing the terms of order higher, m ¨ x = F ( x , ˙ x ) + ε (cid:16) ∂ F ∂ x ˙ x + 1 m ∂ F ∂ ˙ x F (cid:17) , (11)where we have replaced ¨ x again by its approximation (10). This equationdoes not have the problem of runaways and therefore of the choice of initialdata.Using a third order Runge–Kutta methods, we integrate this equation,with the Gaussian vector potential as given by (8), for several values ofthe intensity A . Figure 5 show the orbits found: they are computed byfirst integrating the Abraham–Lorentz–Dirac equation backward for a certainamount of time, and then the Landau–Lifschitz equation forward in time, sothat the initial mechanical data for the two equations agree. One can checkthat the orbits for low values of A essentially coincide, while they differ forhigher field intensities as expected.A more meaningful comparison is given in figure 6, where is reported theloss of energy in a collision versus the energy of the incoming electron, for dif-ferent values of the field intensity, ranging from A = 0 .
05 to A = 0 .
85: in theleft panel the loss is computed from the Landau–Lifschitz, while on the rightthe loss is computed according the Abraham–Lorentz–Dirac equation. Theyare qualitatively different also for field’s intensities well below the threshold,inasmuch as according Abraham–Lorentz–Dirac the loss has a well definedmaximum at a definite energy, while according Landau–Lifschitz it keeps in-creasing without limit, at least in the range of energy we have explored. Inaddition, the energy loss is systematically smaller for the Abraham–Lorentz–Dirac equation with respect to its approximation.
We have show that, for the nonrelativistic Abraham–Lorentz–Dirac equation,there exists a threshold for the intensity of the incoming field, above which15ne has several nonrunaway solutions for the same mechanical initial data ofposition and velocity. Such a threshold agrees well with the bifurcation valueof the main unstable point from saddle to saddle–focus.Above such a threshold the Landau-Lifschitz approximation is clearly indefect, but we have checked that also below threshold the loss of energy ina collision does not agree. As this is the main experimental observable, onemay wonder whether the use of the full Abraham–Lorentz–Dirac equationsinstead of the Landau–Lifschitz approximation might lead to an agreementbetween theory and experiment, better than the one found in ref. [8].Answering such a question would require an analysis similar to that per-formed in this pape, for the full relativistic Abraham–Lorentz–Dirac equa-tion, in the regime of interest for the experiments. This is a much morecomplex task, on which we hope to come back in the future.
Acknowledgments . We thank prof. Galgani for indicating to us the rolethe full Abraham–Lorentz–Dirac equation might present in explaining theexperiments of scattering of electrons by strong laser pulses, and for manydiscussions and suggestions.
References [1] A. Carati, P. Delzanno, L. Galgani and J. Sassarini,
Non uniquenessproperties of the physical solutions of the Lorentz–Dirac equation , NonLinearity , pp 65–79 (1995).[2] M. Abraham, Prinzipien dex Dynamik des Elektrons , Ann. Phys. (1903).[3] H.A. Lorentz, Theory of electrons , second edition (Teubner,Liepzig,1916).[4] P.A.M. Dirac,
Classical theory of radiating electrons , Proc. R. Soc. A , 148–169 (1938).[5] L.D. Landau, E.M. Lifshitz,
The Classical Theory of Fields ,(PergamonPress, Oxford, 1971).[6] A. Di Piazza,
Exact Solution of the Landau-Lifshitz Equation in a PlaneWave , Lett. Math. Phys. , 305 (2008).167] J.M Cole et al., Experimental Evidence of Radiation Reaction in theCollision of a High-Intensity Laser Pulse with a Laser-Wakefield Accel-erated Electron Beam , Phys. Rev. X , 011020 (2018)[8] K. Poder et al., Experimental Signatures of the Quantum Nature ofRadiation Reaction in the Field of an Ultra intense Laser , Phys. Rev. X8