Exact solutions in radiation reaction and the radiation-free direction
EExact solutions in radiation reaction and the radiation-free direction
Robin Ekman, ∗ Tom Heinzl, † and Anton Ilderton ‡ Centre for Mathematical Sciences, University of Plymouth, Plymouth, PL4 8AA, UK
We present new exact solutions of the Landau-Lifshitz and higher-order Landau-Lifshitz equationsdescribing particle motion, with radiation reaction, in intense electromagnetic fields. Through thesesolutions and others we compare the phenomenological predictions of different equations in thecontext of the conjectured ‘radiation-free direction’ (RFD). We confirm analytically in several casesthat particle orbits predicted by the Landau-Lifshitz equation indeed approach the RFD at extremeintensities, and give time-resolved signals of this behaviour in radiation spectra.
I. INTRODUCTION
Despite having been studied for more than a century [1–3], radiation reaction (RR) continues to attract theoreti-cal [4–9], computational [10, 11], and experimental [12–15]interest. In large part, this attention is driven by intenselaser systems [16–19] now granting access to regimes wherequantum and classical RR forces can dominate the Lorentzforce. For recent reviews see Refs. [20, 21].It is well-known that including RR effects allows fornew phenomena, such as anomalous particle trapping [22],chaotic motion [23], symmetry breaking [24, 25], andsignificantly enhanced generation of certain plasma wavemodes [26]. In such scenarios it is often expected that clas-sical RR effects receive significant quantum corrections;the simpler setting of classical physics can, nevertheless,still provide important insight [27], and classical effectscan persist in quantum theory [22]. From a theoreticalpoint of view, classical radiation reaction also remains aninteresting test-bed for the emergence of non-perturbativephysics [3, 7, 27, 28].Eliminating the radiation fields, created by the charges,from the classical equations of motion, one arrives atthe Lorentz-Abraham-Dirac (LAD) equation [1–3] whichcontains the third time derivative of position. This im-plies unwanted effects such as runaway solutions andpre-acceleration. The Landau-Lifshitz [29] (LL) equation,obtained from LAD through ‘reduction of order’ (see be-low) is however free of these difficulties and is typicallyan excellent approximation below scales where quantumeffects appear [5].Given the subtlety of unphysical non-perturbative ef-fects, and the number of complex phenomena attributableto RR, exact solutions can help in making precise state-ments. In this paper we present two new exact solutionsfor field configurations that depend only on a single light-like direction, with polarisation either longitudinal ortransverse to that direction. Specifically, for the longitu-dinal case we give the exact solution of the LL equation;and for transverse polarisation, i.e. plane waves, we solve ∗ [email protected] † [email protected] ‡ [email protected] the second-order Landau-Lifshitz equation obtained by it-eration of reduction of order in closed form, in a physicallymotivated setup.As an application of these exact solutions (and othersto be discussed), we provide analytic evidence confirmingthe tendency of radiation reaction in strong fields toalign particle motion with the ‘radiation free direction’(RFD) in which they locally experience zero accelerationtransverse to their direction of motion, thus minimisingradiation losses. The RFD hypothesis was suggested andsupported by numerical simulations in [30].This paper is organised as follows. We first give nota-tion and conventions, and write down the Lorentz force,LAD and LL equations for reference. In Sect. II we solvethe LL equation in longitudinally polarised electric fieldsof arbitrary strength and form, and demonstrate analyt-ically that orbits transition to the RFD. We also showthat RFD dynamics distinguishes between other proposedclassical equations of motion. In Sect. III we investigateRFD dynamics in the case of plane wave, i.e transverselypolarised backgrounds, for which the solution of the LLequation is already known, and look for signals of (thetransition to) RFD dynamics in emission spectra. InSect. IV we solve the ‘second-order’ LL-like equation, ob-tained from iteration of reduction of order, in plane wavesand compare its predictions with those of the standardLL equation. We conclude in Sect. V.
A. Notation and conventions
We use Heaviside-Lorentz units with c = (cid:126) = 1 andemploy lightfront coordinates x ± := x ± x and x ⊥ :=( x , x ). Lightfront momentum components p ± and p ⊥ are defined analogously. It is convenient to introduce alightlike vector n µ such that x + = n · x .In a background field F µν the LAD equation of particlemotion is obtained from the coupled system of the Lorentzforce law and Maxwell’s equations by integrating out thedynamical electromagnetic fields. Writing f ≡ eF/m , theLAD equation is¨ x µ = f µν ˙ x ν + τ P µν ... x ν , (1)where a dot is a proper-time derivative, τ := e / πm is the characteristic timescale for RR, and P µν projectsorthogonally to ˙ x µ . The standard Lorentz force equation a r X i v : . [ h e p - ph ] F e b for motion in the background only, i.e. neglecting radiationand RR, is recovered by setting τ = 0.The LL equation is obtained by substituting (1) backinto itself to eliminate the third derivative in favour ofnew, explicitly f -dependent terms, and then neglectingterms of order τ and larger. The LL equation thus foundis ¨ x µ = f µν ˙ x ν + τ ˙ f µν ˙ x ν + τ P µν f νρ f ρσ ˙ x σ . (2)This process is called ‘reduction of order’, since it replacesa third-order ODE, the LAD equation with a second-orderODE. The process can of course be extended to higherorders in τ ; we return to this in Sect. IV. II. LONGITUDINAL POLARISATION
We consider first electric fields depending only on x + ,which represent electromagnetic pulses propagating in thenegative z direction. We take the fields to be ‘longitudi-nally’ polarised in the z -direction [31, 32]. This is not asolution of the source-free Maxwell equations, but wavesof this type can be realised in a plasma or using binaryoptics for light [33]. The non-zero components of the fieldtensor are F + − = − F − + = ∂ + A − ( x + ) , (3)equivalent to having a purely electric field E = ( F − + /
2) ˆ z .This class of fields includes of course constant electric,on which we comment below. Recall that any massiveparticle orbit can be parameterised by lightfront time [34].Using this, the solution of the Lorentz force equation inour background is easily found; writing a ( x + ) = eA + , thefirst integrals, i.e. the particle momenta π µ = m ˙ x µ , are(Lorentz) π + ( x + ) = p + − a ( x + ) , (4)(Lorentz) π ⊥ ( x + ) = p ⊥ , (5)with initial conditions π µ = p µ at some initial time x + = x + before the field turns on. π − follows from the mass-shell condition, π + π − − π ⊥ π ⊥ = m , and the momentadetermine the orbit, x µ ( τ ), via quadratures.Observe that we have uniform motion perpendicular tothe pulse direction in (5), for any p ⊥ , as the ‘transverse’directions x ⊥ decouple. According to [30], though, whenRR is included the particle should move toward the RFDwhich, for the field (3), is parallel or anti-parallel tothe electric field polarisation according to the sign ofthe charge, in other words the negative or positive z direction. To see the impact of RR effects, we turn to theLL equation.We consider first the transverse components of (2):¨ x ⊥ = = − τ ˙ x + ˙ x − (2 f + − ) ˙ x ⊥ = − τ (1 + ˙ x ⊥ ˙ x ⊥ )(2 f + − ) ˙ x ⊥ , (6)using the mass-shell condition in the second line. It is clearfrom (6) that ¨ x ⊥ <
0, meaning that that RR effects drive the transverse velocity ˙ x ⊥ monotonically to zero. As aresult, the motion becomes confined to the tz plane, whichconfirms that the particle momentum indeed becomesaligned with the RFD. A complementary argument is tonote that the transverse equation of motion (6) alwayshas the trivial solution ˙ x ⊥ = 0; a stability analysis thenshows that this solution represents an ‘attractor’, meaningany deviation from ˙ x ⊥ = 0 will be killed by virtue of (6).Notably, we find that the system on the attractivesub-manifold, ˙ x ⊥ = 0, is integrable; ˙ x − becomes triviallydetermined from the mass-shell condition, ˙ x − = 1 / ˙ x + ,and (2) reduces to a single equation for ˙ x + , m ¨ x + ( x + ) = − ˙ x + ∂ + a − τ ∂ + a , (7)which can be solved exactly. Changing independent vari-able from proper time to lightfront time, (7) becomes ∂ + π + = − ∂ + a ( x + ) − τ m π + ∂ + a ( x + ) , (8)which is readily solved to yield, for π + ( −∞ ) = p + , π + ( x + ) = e − τ a (cid:48) ( x +) m (cid:16) p + − x + (cid:90) −∞ e τ m a (cid:48) ( y ) a (cid:48) ( y ) d y (cid:17) . (9)As a check, we note that for a constant electric field, a (cid:48)(cid:48) = 0, and p ⊥ = 0, the LL solution (9) and the Lorentzsolution (4) agree, and also solve the LAD equation. Thisis consistent with the literature result that there is ‘noradiation reaction for hyperbolic motion’ [35–37]; there isthough radiation [35, 38, p. 399].With (9) one can make explicit the stability of theRFD solution. Linearising the LL equation in π ⊥ , theequations for π ± are unchanged, (since π ⊥ enters them quadratically ), which allows us to solve (6) as π ⊥ ( x + ) (cid:39) π ⊥ ( x + ) exp (cid:2) − mτ (cid:90) x + x +0 f + − /π + d y (cid:3) , (10)where π + is as in (9); thus deviations of motion from theRFD are exponentially suppressed. A. Example: Sauter pulse
To understand the physics of the exact solution (9) innon-constant fields we turn to an explicit example with achosen field configuration. We consider a Sauter pulse ofwidth 1 /ω and peak field strength determined by a , a (cid:48) ( x + ) = mωa ( ωx + ) (11) a ( x + ) = ma (cid:0) ωx + ) (cid:1) , (12)with positive/negative values of a corresponding to anelectric force on the particle parallel/anti-parallel to thedirection of pulse propagation (which, recall, is the nega-tive z -direction in our conventions). For this field the integral in (9) can be performed an-alytically in terms of the error function with the result,for a > π + ( x + ) = e − ωτ a sech ( ωx + ) (cid:20) p + − m (cid:114) πa ωτ e ωτ a (cid:18) erf (cid:114) ωτ a ωx + + erf (cid:114) ωτ a (cid:19)(cid:21) , (13)while for a <
0, absolute value signs should be insertedunder the square roots and erf should be replaced by erfi.The corresponding Lorentz force results are given simplyby inserting (12) into (4).Note that the solutions of both the LL and Lorentzequations are valid only as long as π + >
0, as is requiredfor massive particles. For π + →
0, the particle is acceler-ated to almost co-propagate with the field, and reachesthe speed of light in finite lightfront time, after whichthe particle ‘leaves the spacetime manifold’ [31, 32]. Notethat this finite lightfront time corresponds to infinite lab-frame, or proper, time. As such, considering both (4)and (9), it is clear that the sign of a is a crucial factorin determining properties of the motion.We consider first the case a <
0, for which the force onthe particle is anti-parallel to the propagation direction;for a head-on collision, this means the particle is acceler-ated in its direction of initial propagation. We define themomentum transfer W by W := π + ( ∞ ) − π + ( −∞ ) = π + ( ∞ ) − p + . (14)Without RR, W Lorentz = m | a | , but according to the LLequation RR reduces the momentum transfer; one findsfrom (13) that W LL (cid:39) m αω for αω | a | m (cid:29) . (15)The behaviour of W as a function of a , for both theLorentz and LL equations of motion, is shown in Fig. 1.We turn to the case a >
0, for which the force on theparticle is parallel to the field propagation direction (andtherefore opposite the initial propagation direction of theparticle). In this case we find that for sufficiently large a ,the particle can be brought to rest, and for even larger a can be caught in the pulse and accelerated almost tothe speed of light. In the Lorentz case, the respectivethresholds above which these phenomena occur are easilyread off from (4) as a , stop = p + /m − a ,c = p + /m . (16)Including RR through the LL equation, the thresholds arethe solutions of an intractable transcendental equation,but it is easily found by numerical investigation that boththresholds are lowered by RR. What this means is thatthere is a range of a such that particles are back-scattered by RR effects, i.e. their direction of motion is reversed.Example orbits illustrating this effect are plotted in Fig. 2.In the figure, an unphysically large value of α has beenused to exaggerate the effect of RR (along with an ω much larger than is phenomenologically motivated). Thisis because the solution (9) describes a particle alreadymoving in the RFD, and because longitudinal forces pro-duce much less radiation than transverse forces, RR isconsequently a very small effect. RR will naturally bemore pronounced in the transition to the RFD, i.e. as π ⊥ is driven to zero and, when close to zero, follows (10).Rather than analyse the radiation spectra from this ap-proximate final stage motion, though, we will considerin the next section the case of plane waves, which admitexact solutions to the LL equation for arbitrary initialconditions. Before doing so we compare the results abovewith those of other classical equations. B. Alternative equations
We ask here what different classical equations predictregarding transition to, and motion in, the RFD. Thequestion is somewhat academic, as unlike LAD (the clas-sical equation of motion) or LL (an approximation to it)
Landau - Lifshitz Lorentz
FIG. 1. Momentum transfer W (14), in a head-on collisionwith a Sauter pulse with a <
0. Without RR, the momentumtransfer is unbounded. With RR, W approaches a constantat large | a | , as RR effects dominate over the electric force. - - - FIG. 2. Particle orbits in a head-on collsion with a longitudi-nally polarised Sauter pulse. Solid: LL, dashed: Lorentz. TheLL orbits have been displaced by unity in the ωx − directionto distinguish them from the Lorentz orbits. Note that anunphysically large α has been used to exaggerate RR. other equations proposed over the years bring in someexternal assumption. We will therefore be brief.Eliezer’s equation [39] (rederived by Ford-O’Connell ina slightly different context [40]) is¨ x µ = f µν ˙ x ν + τ P µν dd τ ( f νρ ˙ x ρ ) . (17)Linearising in the transverse momentum as above, we find˙ π ⊥ (cid:39) (cid:18) τ m ˙ π + π + ∂ + a (cid:33) π ⊥ , π + = π + LL , (18)This means that motion in the RFD, when it is reached, isexactly as described by LL. The approach to the RFD is,however, not monotonic: direct numerical investigationconfirms that the coefficient of π ⊥ in (18) is initiallynegative, but can change sign, implying attraction to orrepulsion from the RFD.Ultimately, it is impossible to turn off quantum effects,and the predictions of some proposed equations do notagree with the classical limit of QED results [41–43]. Anexample is the Mo-Papas equation [44],¨ x µ = f µν ˙ x ν + τ P µν f νρ ¨ x ρ . (19)Linearising again, one finds in this case that˙ π ⊥ (cid:39) − τ m ( ∂ + a ) π ⊥ , π + = π + Lorentz (20)The first of these equations tells us that, for small trans-verse momentum, ˙ π ⊥ = 0 is again a stable attractor, andthe orbit goes to the RFD. However, motion in the RFDis that predicted by the Lorentz force equation, withoutradiation reaction. Thus the LL, Eliezer and Mo-Papas,equations all predict different behaviour, and it seemsthat, in principle, RFD dynamics can differentiate be-tween classical equations of motion. III. TRANSVERSE POLARISATION
We turn now to plane waves, i.e. functions of x + = n · x ,which are transversely, rather than longitudinally po-larised. Their potential a µ := eA µ can always be written a µ = ma δ ⊥ µ f ⊥ ( ωx + ) in which ω is some frequency scale, f ⊥ describes the shape of the field, and the dimensionlessinvariant a characterises the peak field strength.The solution to the LL equation was first given formonochromatic, linearly polarised plane waves in Ref. [45],and extended to all plane waves in Ref. [46]. It is conve-niently presented using the following parameterisation ofthe momentum π µ , π µ = 1 h (cid:18) p µ − B µ + n µ p · B − B + m ( h − n · p (cid:19) , (21)in which B µ is purely transverse. Writing φ ≡ ωx + , k µ ≡ ωn µ and a prime for a φ -derivative, the functions h and B µ are h = 1 + τ k · pm (cid:90) φ −∞ d y a (cid:48) ( y ) · a (cid:48) ( y ) , (22) B µ = τ k · pm a (cid:48) µ + (cid:90) φ −∞ d y h ( y ) a (cid:48) µ ( y ) , (23)with initial conditions π µ ( −∞ ) = p µ . The parame-terisation (21) is chosen because it makes clear that all momentum components are actually proportional to1 /h = π + /p + ; in this sense π + governs the dynamics.It is more convenient to analyse the transition to theRFD at the level of the solution (21), rather than at thelevel of the LL equation, as we did for longitudinallypolarised fields. Using (22)–(23), we see that the ratio oftransverse momentum to z -momentum, | π ⊥ | π z = 2 | π ⊥ | π + − π − , (24)behaves as ∼ a − for a (cid:29)
1. Hence, while motion ina plane wave can never be confined exactly to the tz plane due to the explicitly transverse polarisation, motionis dominantly in the z direction for a (cid:29)
1. This isindeed the radiation-free direction according to Ref. [30],
FIG. 3. Electric field of a circularly polarised pulse with asin envelope, cf. (25).
50. 100. 500. 1000.
FIG. 4. Outgoing scattering angle, relative to the laser prop-agation direction, as a function of a , at fixed initial particleenergy and for several incoming collision angles, indicatedat each solid curve, in the pulse (25). The asymptotic 1 /a scaling is also shown (dashed). (The scattering angle is essen-tially independent of a below the range displayed.) Opticalfrequency, ω = 1 eV, is chosen as being the most phenomeno-logically relevant; the asymptotic behaviour then sets in for an a of several hundred, owing to the smallness of ωτ (cid:39) − . although the limits of the formulae provided there needto be taken with care as, in a plane wave, both fieldinvariants vanish.We illustrate the particle motion using a circularly po-larised, few-cycle pulse with a sin envelope, for which theintegrals in (22) and (23) can be performed analytically.The pulse is given by (cid:18) a (cid:48) a (cid:48) (cid:19) = ma sin φ (cid:18) cos φ sin φ (cid:19) (25)for 0 ≤ φ ≤ π , and vanishing otherwise, see Fig. 3; theenvelope ensures that the electromagnetic fields, and theirderivatives, vanish at the edge of the pulse.Performing the integrals in (22) and (23), we can fi-nesse the approximation (24) and calculate the outgoingscattering angle θ f of the incident particle with respectto the laser propagation direction. We find θ f ≈ παa ω a (cid:29) , (26)which, note, is completely independent of the incom-ing direction of the particle, and is clearly in agreementwith (24). Fig. 4 shows the full a -dependence, and theasymptote (26), for several incidence angles.This confirms that, at high intensity, all particles aredriven to propagate dominantly in the RFD (with rel-atively small transverse momentum). We illustrate thetransition to (near) laser-collinear scattering by plottingparticle orbits in the ( t, z ) plane in Fig. 5. Note that theparameters in Fig. 5 have been chosen for visual clarityrather than physicality: for realistic parameters and a
10 20 30 - - - - - FIG. 5. Particle orbits for a head-on collision with thepulse (25), whose spacetime extent is indicated by the dashedlines. As a increases, the particle goes from scattering for-ward, to coming to a stop, to scattering backward. (Note: anunphysically large α has been used to exaggerate RR.) in the radiation-dominated regime, the particle is carriedwith the pulse for many cycles’s worth of lab time. A. Radiation spectra
Signatures of RFD dynamics can be found in the emit-ted radiation spectra. The radiated photon 4-momentum K µ has the standard form K µ = − (cid:90) d (cid:96) ⊥ d (cid:96) + (2 π ) (cid:96) + (cid:96) µ | j ( (cid:96) ) | , (27)where j µ is the current in Fourier space, j µ ( (cid:96) ) = − e (cid:90) d x + e i(cid:96) · X ( x + ) ∂∂x + (cid:16) π µ i(cid:96) · π (cid:17) , (28)and X µ is the particle orbit. All but one of the integralsin (27) can be performed exactly; the final expression for K µ is K µ = 23 e π k · pm (cid:90) d φ m h (cid:48) ( x ) − B (cid:48) ( x ) m h ( x ) π µ ( φ ) . (29)Examining the integrand shows us from where in the pulseradiation is generated. For low a RR is negligible, andwe find that emission of all spectral components K µ issupported mainly near the peak of the pulse. A signatureof the transition to the RFD is then that emission intodifferent spectral components K µ becomes time-resolved at high intensity; as shown in Fig. 6(a), radiation emittedanti-collinear/transverse/collinear to the laser is predomi-nantly emitted near the beginning/peak/end of the pulse.This corresponds directly to the change in particle direc-tion associated with transition to the RFD, as sketchedin Fig. 6(b).In Fig. 6(a) the components are normalised to theirmaxima to highlight their different temporal supports.To extract analytic estimates for their relative sizes, when a (cid:29)
1, requires a little care. The components K + , K − scale according to naive estimates using (21), viz., K ⊥ = O ( a ) and K − = O (cid:0) a (cid:1) . (30)However, the dominant contribution to K + does not scaleas a − as might be expected; the dominant contributioncomes from a boundary term hidden in the contributionfrom B (cid:48) and is [27] K + p + ∼ − e π k · pm (cid:90) d φ h a (cid:48) h = − (cid:90) d φ h (cid:48) h = 1 . (31)This scaling behaviour is a consequence of conservationof total lightfront momentum; K + and π + both beingnon-negative, the radiation field cannot carry off morethan the initial p + of the particle.While the total perpendicular momentum is also con-served, π ⊥ is not constrained and can always absorb anarbitrarily large recoil. Finally, due to the lack of symme-try in x + the collinear momentum is simply not conserved,putting no restriction on the particle “pumping” lasermomentum into its own radiation field.It may seem paradoxical that, comparing (31) and (30),the dominant component is in the RFD, but we mustremember that motion in the RFD minimises radiation for a given magnitude of the applied force [30]. In theradiation-dominated regime, though, the particle sheds allof its lightfront momentum early in the rise of the pulse.As the particle approaches the radiation-free direction themagnitude of the force and, eventually, the Lorentz factorincrease, both of which strongly enhance the radiatedpower. IV. LL TO SECOND ORDER
The LL equation (2) has been obtained by iterating theLAD equation (1) to first order in τ . To assess the qualityof this approximation we consider the size and influenceof corrections by performing the iteration to second orderin τ . We will refer to the LAD equation iterated to n :th order in τ as LL n (so that the Lorentz equation ofmotion is LL ). A somewhat lengthy calculation yieldsthe following expression for LL ,¨ x µ = (cid:0) f + τ ˙ f + τ ¨ f − τ ( ˙ xf ˙ x ) f (cid:1) µν ˙ x ν + τ P µν (cid:2) f νρ + 2 τ f νρ + 2 τ ( f ) • νρ (cid:3) ˙ x ρ . (32) A. Transverse polarisation
As for LL , if one can solve LL for π + in a plane wave,as a function of x + , then the transverse momenta π ⊥ (a) ˆ x ˆ z (b) FIG. 6. Radiation reaction makes the radiation profile in apulse time-resolved. (a): The radiated 4-momentum in thepulse (25), as a function of lightfront time. Each componenthas been normalised to its maximum, ˜ K µ = K µ / max | K µ | ,and the dashed curves have been reflected in the horizontalaxis for clarity. (b): Sketch of the particle orbit and theradiation cone rotating along it. are easily calculated, and π − is given by the mass-sellcondition. We therefore focus on π + .The momentum π µ is again conveniently parameterisedas in (21), so that we again have π + /p + = 1 /h . It isuseful to define the two dimensionless RR parameters δ = 23 e π k · pm , ∆ := a δ , (33)where δ is essentially an energy parameter, which shouldstrictly be small in the classical regime, whereas ∆ de-pends on the field strength, and so can be large. Denoting φ -derivatives with a prime as before, the LL equationfor h becomes h (cid:48) = − δm a (cid:48) · a (cid:48) − δ m h a (cid:48) · a (cid:48)(cid:48) , (34)which is nonlinear due to the factor of h − in the finalterm: this is an Abel equation of the second kind, andthus not analytically solvable in general, while specialcases are usually only solvable parametrically [47].However, there is a solvable case which is relevant tothe physical situation of interest. We have seen that athigh intensities, a particle can be stopped and turnedaround soon after it enters the pulse, i.e. before reachingthe peak. We therefore consider dynamics in the initial rise of the pulse, a simple model of which is a (cid:48) · a (cid:48) = − m a e φ , φ < . (35)The individual components of a (cid:48) may contain oscillatoryfactors, c.f. the brackets in (25), without affecting (35).(Simple extensions are to continue directly to φ > e −| φ | tomodel a sharply peaked field [48, 49].) With this choice,we bring (34) to standard Abel form by changing theindependent variable from φ to z := 1 − δ φ (cid:90) −∞ d y a (cid:48) ( y ) · a (cid:48) ( y ) = 1 + ∆ e φ , (36)which, note, is just the solution for h from LL . OurODE (34) reduces to h ( z ) ∂ z h ( z ) − h ( z ) = 2 δ . (37)Setting all explicit factors of δ → solution; this implies, as suggested inRef. [50] that the difference between LL and LL dependsessentially on δ (which should be small in the classicalregime), rather than the potentially large ∆. To confirmthis, we need the solution of (37). This is easily foundby first solving the equation for z as a function of h .Choosing initial condition h ( z = 1) = 1, the solution is z = h − δ log h + 2 δ δ . (38)Again, setting δ = 0 on the right correctly recovers theLL result: the difference is explicitly dependent only on δ . Rearranging, we can write h in terms of the Lambertfunction [51], or product logarithm, W , − h ( φ )2 δ = 1+ W (cid:18) − δ δ exp (cid:20) − − e φ δ (cid:21)(cid:19) . (39)Interestingly, the argument of W is in the range ( − /e, W ≡ W − , i.e. not theprincipal branch of W . As is clear from the expressionsabove, and as shown explicitly in Fig. 7, the differencebetween h as predicted by LL vs. LL is extremely smallunless the energy parameter δ is taken to be very large,and therefore outside the classical regime. B. Longitudinal polarisation
For the longitudinally polarised field, (32) implies thefollowing LL equation for the transverse velocity compo-nents,˙ u ⊥ = − τ u + u − (cid:2) (1 + 2 τ u + ∂ + )( f + − ) (cid:3) u ⊥ . (40) - - - - - - ϕ / h δ = / Δ = δ = / Δ = δ = Δ = FIG. 7. Momentum π + /p + ≡ /h in the rise of the pulse,for various δ and ∆, in LL (solid lines) and LL (dashedlines). Increasing the field strength, so ∆, leads to significantmomentum loss earlier in the pulse. The rate of this loss iscorrected by effects of O ( δ ) in LL . Comparing to (6), the difference is the derivative terminside the brackets; this does not have a definite sign, andso the approach to the RFD is no longer uniform. However,this new NLO term is roughly of order αωp + /m relativeto the LO term, so is again dependent on energy ratherthan field strength. As such, higher-order corrections areonly significant when the energy parameter is large. Weconclude therefore that the RFD hypothesis continues tohold (at least) for parameters where a classical treatmentis valid. V. CONCLUSIONS
Exact solutions of equations of motion allow us to makestatements that do not rely on approximations, and cangive explicit insight into complex phenomena induced by,as considered here, radiation reaction. We have presentednew exact solutions of the Landau-Lifshitz equation, fora longitudinally polarised electric field depending on asingle lightlike coordinate, and of the ‘second-order-of-reduction’ Landau-Lifshitz equation (LL ) for a planewave.We have used these solutions to examine the radiation-free direction hypothesis [30], that is the proposed uni-versal approach of particle motion, in strong fields, toa direction which minimises radiation losses. We haveexplicitly confirmed this behaviour in our new solutions.Many authors have found exact solutions to the LADand/or LL equations in a number of field configura-tions, including fields depending only on time [28], con-stant electromagnetic fields [36, 52, 53], rotating elec-tric fields [5, 53, 54], the Coulomb potential in the non-relativistic limit [55], and plane waves [45, 46]. Theseworks, the majority of which predate the RFD hypothesis,contain implicit support for it. For example, in a rotatingelectric field E = E (cos ωt, sin ωt, e E byan angle ≈ ◦ for small E and ∼ E − / for large E [5,Sec. II.A]. Further, although the exact solution of LAD isnot known even in a general constant field, its asymptoticbehaviour is known [56], namely the particle worldlinebecomes confined to an eigen-2-plane of the field tensor;the same holds for the exact solution of LL [36].There are many examples of particle motion for whichthe Lorentz force law is integrable, or even superinte-grable [57–59]. In future work it would be interesting toexamine in detail how this integrability is affected by theaddition of radiation-reaction terms in the LAD, LL andhigher order, LL n , forms. For example, the Lorentz forceequation in a plane wave is superintegrable; in going tothe Landau-Lifshitz equation (LL ), one loses conservedquantities (i.e. π + is no longer conserved), which lowersthe degree of integrability, while the second-order Landau- Lifshitz equation (LL ) becomes nonlinear and is onlyintegrable in special cases.Finding exact solutions of higher-reduction-of-orderequations could shed light on the emergence of non-perturbative phenomena (runaways and acausal solu-tions), but this would require resummation [7]. This is par-ticularly intriguing as resummation has recently been high-lighted as being essential for fully understanding the be-haviour of quantum dynamics in strong fields [27, 60, 61]. ACKNOWLEDGMENTS
The authors thank Ben King for useful comments anddiscussions. The authors are supported by the LeverhulmeTrust (RE, AI, TH), grant RPG-2019-148, and the EP-SRC (AI), grant EP/S010319/1. [1] M. Abraham,
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