Exact solution of the Landau Lifshitz equations for a radiating charged particle in the Coulomb potential

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1 Available online at Annals of Physics 323 (2008) Exact solution of the Lanau Lifshitz equations for a raiating charge particle in the Coulomb potential S.G. Rajeev Department of Physics an Astronomy, University of Rochester, Rochester, NY14627, USA Receive 7 January 2008; accepte 12 January 2008 Available online 2 February 2008 Abstract We solve exactly the classical non-relativistic Lanau Lifshitz equations of motion for a charge particle moving in a Coulomb potential, incluing raiation amping. The general solution involves the Painlevè transcenent of type II. It confirms our physical intuition that a negatively charge classical particle will spiral into the nucleus, supporting the valiity of the Lanau Lifshitz equation. Ó 2008 Elsevier Inc. All rights reserve. PACS: m; Ap Keywors: Raiation reaction; Lanau Lifshitz equation; Lorentz Dirac equation; Coulomb fiel 1. Introuction A corner stone of theoretical physics is the exact solution of the motion of a particle moving in an inverse square law force. The orbits are conic sections: ellipses or hyperbolae epening on initial conitions. The original application was to the motion of planets aroun the Sun [1]. Later the same problem was foun to arise in the Rutherfor scattering of alpha particles an in the classical moel of the atom. The orbits of charge particles in a Coulomb potential cannot be conics exactly, as it is a funamental principle of electroynamics that all charge particles must raiate when aress: rajeev@pas.rochester.eu /$ - see front matter Ó 2008 Elsevier Inc. All rights reserve. oi: /j.aop

2 S.G. Rajeev / Annals of Physics 323 (2008) accelerate [2,3]. The raiation carries away energy, an therefore acts as a issipative force, changing the equation of motion. Can we still fin an exact solution for the motion of the particle in a Coulomb fiel, after taking account of raiation reaction? Deriving the correct equation of motion for a charge particle, incluing this raiation amping, is not a simple matter. The problem is that raiation reaction is the force exerte on the particle by its own electromagnetic fiel; a straightforwar application of the Lorentz force law will give an infinite force in the case of a point particle. Dirac [4] foun a way through this minefiel of ivergences an euce an equation of motion incluing the raiation reaction. A key point was that the ivergences are remove by a renormalization of the mass of the charge particle. This Lorentz Dirac equation of motion is a thir orer ODE, as the raiation reaction force is proportional to the erivative of acceleration. Typical initial conitions will give unphysical solutions that runaway : the energy grows without boun instea of ecaying. Thus Dirac s work, although a major step forwar, cannot be the final wor on the equation of motion of charge particles. Spohn [3,5] showe that these unphysical runaway solutions can be avoie if the initial ata lie in a critical manifol ; i.e., if the initial conitions are fine-tune to avoi the runaway unphysical solutions. This turns out to be the same as treating the force ue to the raiation as a first orer correction. We get this way a secon orer equation with physically sensible solutions. Although without the moern unerstaning, this equation of motion for a raiating particle were given first in the classic text of Lanau an Lifshitz [6]. Therefore these are known now as Lanau Lifshitz (LL) equations of motion. See Ref. [7] for a physical argument in support of the LL equations. There is no unanimity yet that these are the exact equations of motion of a raiating charge particle[8]. In aition to experimental tests, we nee to verify their theoretical consistency. As an example, it shoul not be possible for a negatively charge particle to orbit a nucleus in an elliptical orbit: it shoul plunge into the nucleus as the raiation it emits carries away energy an angular momentum. It is important to verify the physical correctness of the LL equations by checking that its solutions have this property. In this paper, we solve exactly the non-relativistic Lanau Lifshitz equations in a Coulomb fiel; the general solution is in terms of [9,10] Painleve transcenents of type II. The same ifferential equation, with ifferent initial conitions, appears in several other physical problems, such as the Tracy-Wiom law for ranom matrix eigenvalues [11,12]. Our solution turns out to have the correct asymptotic properties: the orbit of a negatively charge particle oes spiral in towars the nucleus. The Lorentz Dirac equation of motion has been stuie in the Coulomb fiel. It has a complicate, unphysical behavior, an no general exact solution is known. For example, in attractive Coulomb potential there are solutions that accelerate away to infinity. See Ref. [3] Section 6 15, [13] The an approximate treatment of the raiative reaction is closer to our physical results [14]. Due to quantum effects, the LL equations cannot be the right escription at the short istances characteristic of atoms. Our solution might still be an approximate escription of an electron with a large principal quantum number in an atom. It shoul also escribe an alpha particle scattere by a nucleus an an electron (or positron) emitte by a nucleus, all of whose motion is affecte by the raiation emitte. Relativistic corrections will become important as the velocity of the particle grows; we are currently investigating the exact solvability of the relativistic LL equations in a Coulomb fiel.

3 2656 S.G. Rajeev / Annals of Physics 323 (2008) The LL equations have alreay been solve in a constant electric fiel an in a constant magnetic fiel [15]. We hope that more physically realistic situations will open up to stuy using the techniques we escribe here. In another paper [16] we have propose a canonical formulation an a quantum wave equation for issipative systems of a particular type. The case we stuy here happens to be of this type, so we hope that a quantum treatment of a raiating electron in an atom along these lines is also possible. This might allow us to go beyon perturbation theory in the calculation of line-withs of a hyrogenic atom. The analogous problem in General Relativity of a star being capture by a blackhole, its energy an angular momentum being carrie away by gravitational raiation is of great importance in connection with the LIGO project to etect gravitational waves. We hope that our solution of the much simpler electroynamic problem will help in unerstaning this case as well. 2. The Lanau Lifshitz equations The LL equation of motion of a raiating charge particle in an electrostatic fiel is [6], " ( )# v a c2 ½cvŠ ¼a þ s cðv rþaþ a t c 2 c v ðv aþ2 2 a2 ; ð1þ c 2 where Also, a ¼ q m E; c ¼ 1 q ffiffiffiffiffiffiffiffiffiffiffi : ð2þ 1 v2 c 2 s ¼ 2 q 2 3 mc ; ð3þ 3 q an m being the charge an mass of the particle, respectively. The issipation parameter s has units of time; for the electron it woul be the (2/3) of the classical electron raius ivie by c. In the non-relativistic limit, it is much simpler: ½v þ srušþru ¼ 0; t v where U is q m ru ¼ ^ru r ; U r ¼ U r ; t ½v þ s^ru ršþ^ru r ¼ 0: ¼ r t times the electrostatic potential. For a central potential, Taking the cross prouct with the position vector gives, with L ¼ r v, L t ¼ s U r r L: ð7þ Thus the irection of angular momentum is preserve. If the initial conitions are such that L 6¼ 0, the motion will lie in the plane normal to this vector. If L ¼ 0 initially, it re- ð4þ ð5þ ð6þ

4 mains zero an the motion is along a straight-line passing through the center of the potential. Using the stanar ientities v 2 ¼ v 2 r þ L2 r ; 2 v r ¼ r t ¼ ^r:v; v r t ¼ L2 r v þ ^r: 3 t we get the system of ODE v r ¼ r t ; 3. The Coulomb potential t ^r ¼ 1 r ½v v r^rš; ^r: ^r ¼ 0; ð9þ t t ½v r þ su r Š¼ L2 r 3 U r; L t ¼ s U r r L: For the Coulomb potential U ¼ k ; U r r ¼ k an r 2 L t ¼ ksl r : ð12þ 3 Thus the centrifugal force is a total time erivative: L 2 r ¼ 1 3 2ks t L2 : ð13þ This coincience allows to write the raial force equation as Put t v r s r 2 1 2ks L2 ¼ k r : 2 z ¼ v r ks r 1 2 2ks L2 ð15þ to write this as L t ¼ ks r L; z 3 t ¼ k r ; r 2 t ¼ L2 2ks þ z þ ks r : ð16þ 2 Using the fact that this is an autonomous system (i.e., t oes not appear explicitly) we can eliminate t, to get a system of two ODEs, L z ¼ s r L; S.G. Rajeev / Annals of Physics 323 (2008) r z ¼ 1 2k 2 s 2 r2 L 2 þ 1 k r2 z þ s: We note as an asie that in the case of purely raial motion, L ¼ 0 this reuces to a Riccati equation for q ¼ 1: r q h z ¼ z i k þ sq2 : ð18þ This can solve in terms of Airy functions. ð8þ ð10þ ð11þ ð14þ ð17þ

5 2658 S.G. Rajeev / Annals of Physics 323 (2008) Returning to the general case, we can rewrite the above system as a single secon orer ODE: 2 L z ¼ 1 2 2k 2 L3 s k zl: ð19þ Up to scaling, this is the particular case with a ¼ 0 of the Painleve II equation (See Ref. [9], page 345) 2 u x ¼ 2 2u3 þ xu þ a: ð20þ Defining constants h b ¼ s i1 3 pffiffiffiffiffiffiffiffiffiffi ; a ¼ 2k 2 b: ð21þ k we have the solution L ¼ auðbzþ: When k < 0, as for an attractive Coulomb potential, u is purely imaginary an the inepenent variable x ¼ bz is real. u also epens on a complex moular parameter s that is etermine by the initial conitions [10]. The ientities L ¼ r 2 h t ; z t ¼ k r 2 allow us to etermine the polar angle: h z ¼ 1 k L: ð24þ By a quarature an a ifferentiation of the Painleve transcenent, r an h are both foun as functions of the parameter z, etermining the orbit: rðzþ ¼s uðzþ ; hðzþ ¼h uðzþ 1 þ a Z z uðbzþz: ð25þ k z z 1 ð22þ ð23þ 4. Examples 4.1. A ecaying orbit To plot orbits, another form of the equations is sometimes more convenient. Define y ¼ L 2 an change to h as the inepenent variable: 1 r L h ¼ ks r ; h ¼ L 2ks z ks L Lr ; 2 L 2 h ¼ 2ks L r ; L ¼ L2 h r 2ks z to get the thir orer ODE: ð26þ ð27þ

6 S.G. Rajeev / Annals of Physics 323 (2008) y x Fig. 1. A ecaying orbit. 3 y h þ y 3 h þ 2k2 s p ¼ 0: y The orbit is then given by 1 rðhþ ¼ 1 p y ks h ð28þ ð29þ We can fin the orbit by numerically integrating the above thir orer ODE. Let us look at the particular case of the attractive Coulomb potential in more etail. Normal units can be restore by imensional analysis. Numerical solution of (28) allows us to plot a slowly ecaying orbit (see Fig. 1) A capture orbit We want a solution 1 for which r ¼ u z > 0. Note that < 0. If rðzþ vanishes it must be a u0 t simple zero, since near a zero r 1. Thus r z z z 0, L Cðz z 0 Þ where C an z 0 are real constants of integration. If a particle approaches from infinity with angular momentum L 1 an raial velocity v 1, z 1 ¼ v 1 þ 1 2 L2 1 : ð30þ 1 For simplicity, we use in this section the natural units of the problem, with jkj ¼s ¼ 1. Normal units can be restore by imensional analysis.

7 2660 S.G. Rajeev / Annals of Physics 323 (2008) Fig. 2. A capture orbit. Since r z v 1r 2, we get r 1 v 1 ðz z 1 Þ : ð31þ Thus z 1 is a simple zero of u0ðzþ. In other wors, to escribe the capture of an incoming charge particle by an attractive Coulomb potential, we just have to solve the Painleve II equation with the initial conitions z 1 ¼ v 1 þ 1 2 L2 1 ; uðz 1Þ¼p 1 ffiffiffiffiffiffi L 1 ; u0ðz 1 Þ¼0 ð32þ 2 an evolve to a point z 0 < z 1 at which uðz 0 Þ¼0. The complete orbit correspons to the finite range z 0 6 z 6 z 1 in the parameter z. We plot an example, obtaine by numerical calculations, of such a capture orbit below (Fig. 2). The velocity blows up as 1 r 2 for small r. From a finite istance the particle is capture in a finite time. Other cases can be worke out similarly. The formulation of the Painlevè equation in terms of the Riemann-Hilbert problem an Isomonoromy [10,12] give powerful techniques to stuy our solution. In particular, there are a pair of conserve quantities that replace energy an angular momentum in this issipative but integrable system. Also, we can generalize the Rutherfor formula for Coulomb scattering to inclue raiation. We hope to return to such a etaile analysis in a longer paper. Acknowlegments I thank J. Golen, S. Iyer an A. Joran for iscussions. This work was supporte in part by the Department of Energy uner the contract No. DE-FG02-91ER40685.

8 S.G. Rajeev / Annals of Physics 323 (2008) References [1] I. Newton, translate by I.B. Cohen, A. Whitman, The Principia: Mathematical Principles of Natural Philosophy, University of California Press, [2] J.D. Jackson, Classical Electroynamics, thir e., Wiley, [3] F. Rohrlich, Classical Charge Particles, secon e., Aison-Wesley, Reaing, MA, [4] P.A.M. Dirac, Proc. Roy. Soc. Lon. A167 (1938) 148. [5] H. Spohn, Dynamics of charge particles an their raiation fiel. arxiv:math-ph/ (unpublishe); M. Kuze, H. Spohn, SIAM J. Math. Anal. 32 (2000) [6] L.D. Lanau, E.M. Lifshitz, Classical Theory of Fiels, Section 76, Butterworth-Heinemann, [7] F. Rohrlich, Am. J. Phys. 68 (2000) [8] See for example, G.A. e Parga, R. Mares, S. Dominguez, Ann. Fon. L. e Broglie 30 (2005) 283. [9] E.L. Ince, Orinary Differential Equations, Dover, [10] A.S. Fokas, A.R. Its, A.A. Kapaev, V. Yu. Novokshenov, Painlevè Transcenents: The Riemann-Hilbert Approach AMS, Provience, RI (2006). [11] C.A. Tracy, H. Wiom, Comm. Math. Phys. 159 (1994) [12] P. Deift, X. Zhou, Asymptotics for the Painlev II equation, Comm. Pure Appl. Math. 48 (1995) 277. [13] J. Huschilt, W.E. Baylis, Phys. Rev. D17 (1978) 985. [14] C.E. Aguiar, F.A. Barone arxiv/physics/ [15] J.C. Herrera, Phys. Rev. D15 (1977) [16] S.G. Rajeev, Ann. Phys. 322 (2007) ; arxiv:quant-ph/