rajeev 2008

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Annals of Physics 323 (2008) 2654–2661

www.elsevier.com/locate/aop

Exact solution of the Landau–Lifshitz equationsfor a radiating charged particle in

the Coulomb potential

S.G. Rajeev

Department of Physics and Astronomy, University of Rochester, Rochester, NY14627, USA

Received 7 January 2008; accepted 12 January 2008Available online 2 February 2008

Abstract

We solve exactly the classical non-relativistic Landau–Lifshitz equations of motion for a chargedparticle moving in a Coulomb potential, including radiation damping. The general solution involvesthe Painleve transcendent of type II. It confirms our physical intuition that a negatively charged clas-sical particle will spiral into the nucleus, supporting the validity of the Landau–Lifshitz equation.� 2008 Elsevier Inc. All rights reserved.

PACS: 41.60.-m; 41.60.Ap

Keywords: Radiation reaction; Landau–Lifshitz equation; Lorentz–Dirac equation; Coulomb field

1. Introduction

A corner stone of theoretical physics is the exact solution of the motion of a particlemoving in an inverse square law force. The orbits are conic sections: ellipses or hyperbolaedepending on initial conditions. The original application was to the motion of planetsaround the Sun [1]. Later the same problem was found to arise in the Rutherford scatter-ing of alpha particles and in the classical model of the atom.

The orbits of charged particles in a Coulomb potential cannot be conics exactly, as it isa fundamental principle of electrodynamics that all charged particles must radiate when

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accelerated [2,3]. The radiation carries away energy, and therefore acts as a dissipativeforce, changing the equation of motion. Can we still find an exact solution for the motionof the particle in a Coulomb field, after taking account of radiation reaction?

Deriving the correct equation of motion for a charged particle, including this radiationdamping, is not a simple matter. The problem is that radiation reaction is the force exertedon the particle by its own electromagnetic field; a straightforward application of the Lor-entz force law will give an infinite force in the case of a point particle. Dirac [4] found away through this minefield of divergences and deduced an equation of motion includingthe radiation reaction. A key point was that the divergences are removed by a renormal-ization of the mass of the charged particle.

This Lorentz–Dirac equation of motion is a third order ODE, as the radiation reactionforce is proportional to the derivative of acceleration. Typical initial conditions will giveunphysical solutions that ‘runaway’: the energy grows without bound instead of decaying.Thus Dirac’s work, although a major step forward, cannot be the final word on the equa-tion of motion of charged particles.

Spohn [3,5] showed that these unphysical runaway solutions can be avoided if the initialdata lie in a ‘critical manifold’; i.e., if the initial conditions are fine-tuned to avoid the run-away unphysical solutions. This turns out to be the same as treating the force due to theradiation as a first order correction. We get this way a second order equation with phys-ically sensible solutions. Although without the modern understanding, this equation ofmotion for a radiating particle were given first in the classic text of Landau and Lifshitz[6]. Therefore these are known now as Landau–Lifshitz (LL) equations of motion. SeeRef. [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 radiatingcharged particle[8]. In addition to experimental tests, we need to verify their theoreticalconsistency. As an example, it should not be possible for a negatively charged particleto orbit a nucleus in an elliptical orbit: it should plunge into the nucleus as the radiationit emits carries away energy and angular momentum. It is important to verify the physicalcorrectness of the LL equations by checking that its solutions have this property.

In this paper, we solve exactly the non-relativistic Landau–Lifshitz equations in a Cou-

lomb field; the general solution is in terms of [9,10] Painleve transcendents of type II.The same differential equation, with different initial conditions, appears in several otherphysical problems, such as the Tracy-Widom law for random matrix eigenvalues[11,12]. Our solution turns out to have the correct asymptotic properties: the orbit of anegatively charged particle does spiral in towards the nucleus.

The Lorentz–Dirac equation of motion has been studied in the Coulomb field. It has acomplicated, unphysical behavior, and no general exact solution is known. For example,in attractive Coulomb potential there are solutions that accelerate away to infinity. SeeRef. [3] Section 6–15, [13] The an approximate treatment of the radiative reaction is closerto our physical results [14].

Due to quantum effects, the LL equations cannot be the right description at the shortdistances characteristic of atoms. Our solution might still be an approximate description ofan electron with a large principal quantum number in an atom. It should also describe analpha particle scattered by a nucleus and an electron (or positron) emitted by a nucleus, allof whose motion is affected by the radiation emitted. Relativistic corrections will becomeimportant as the velocity of the particle grows; we are currently investigating the exactsolvability of the relativistic LL equations in a Coulomb field.

2656 S.G. Rajeev / Annals of Physics 323 (2008) 2654–2661

The LL equations have already been solved in a constant electric field and in a constantmagnetic field [15]. We hope that more physically realistic situations will open up to studyusing the techniques we describe here. In another paper [16] we have proposed a canonicalformulation and a quantum wave equation for dissipative systems of a particular type.The case we study here happens to be of this type, so we hope that a quantum treatmentof a radiating electron in an atom along these lines is also possible. This might allow us togo beyond perturbation theory in the calculation of line-widths of a hydrogenic atom.

The analogous problem in General Relativity of a star being captured by a blackhole,its energy and angular momentum being carried away by gravitational radiation is of greatimportance in connection with the LIGO project to detect gravitational waves. We hopethat our solution of the much simpler electrodynamic problem will help in understandingthis case as well.

2. The Landau–Lifshitz equations

The LL equation of motion of a radiating charged particle in an electrostatic field is [6],

d

dt½cv� ¼ aþ s cðv � rÞaþ v � a

c2a� c2

c2v a2 � ðv � aÞ

2

c2

( )" #; ð1Þ

where

a ¼ qm

E; c ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffi1� v2

c2

q : ð2Þ

Also,

s ¼ 2

3

q2

mc3; ð3Þ

q and m being the charge and mass of the particle, respectively. The dissipation parameters has units of time; for the electron it would be the (2/3) of the classical electron radiusdivided by c.

In the non-relativistic limit, it is much simpler:

d

dt½vþ srU � þ rU ¼ 0; v ¼ dr

dtð4Þ

where U is qm times the electrostatic potential. For a central potential,

rU ¼ rUr; U r ¼dUdr

; ð5Þ

d

dt½vþ srUr� þ rUr ¼ 0: ð6Þ

Taking the cross product with the position vector gives, with L ¼ r� v,

dL

dt¼ �s

U r

rL: ð7Þ

Thus the direction of angular momentum is preserved. If the initial conditions are suchthat L 6¼ 0, the motion will lie in the plane normal to this vector. If L ¼ 0 initially, it re-


mains zero and the motion is along a straight-line passing through the center of the poten-tial.

Using the standard identities

v2 ¼ v2r þ

L2

r2; ð8Þ

vr ¼drdt¼ r:v;

d

dtr ¼ 1

r½v� vr r�; r:

d

dtr ¼ 0; ð9Þ

dvr

dt¼ L2

r3þ r:

dv

dtð10Þ

we get the system of ODE

vr ¼drdt;

d

dt½vr þ sU r� ¼

L2

r3� U r;

dLdt¼ �s

U r

rL: ð11Þ

3. The Coulomb potential

For the Coulomb potential U ¼ kr ;U r ¼ � k

r2 and

dLdt¼ ksL

r3: ð12Þ

Thus the centrifugal force is a total time derivative:

L2

r3¼ 1

2ksd

dtL2: ð13Þ

This coincidence allows to write the radial force equation as

d

dtvr �

sr2� 1

2ksL2

� �¼ k

r2: ð14Þ

Put

z ¼ vr �ksr2� 1

2ksL2 ð15Þ

to write this as

dLdt¼ ks

r3L;

dzdt¼ k

r2;

drdt¼ L2

2ksþ zþ ks

r2: ð16Þ

Using the fact that this is an autonomous system (i.e., t does not appear explicitly) we caneliminate dt, to get a system of two ODEs,

dLdz¼ s

rL;

drdz¼ 1

2k2s2r2L2 þ 1

kr2zþ s: ð17Þ

We note as an aside that in the case of purely radial motion, L ¼ 0 this reduces to a Riccatiequation for q ¼ 1

r:

dqdz¼ � z

kþ sq2

h i: ð18Þ

This can solved in terms of Airy functions.


Returning to the general case, we can rewrite the above system as a single second orderODE:

d2Ldz2¼ � 1

2k2L3 � s

kzL: ð19Þ

Up to scaling, this is the particular case with a ¼ 0 of the Painleve II equation (See Ref. [9],page 345)

d2udx2¼ 2u3 þ xuþ a: ð20Þ

Defining constants

b ¼ � sk

h i13

; a ¼ffiffiffiffiffiffiffiffiffiffiffi�2k2

pb: ð21Þ

we have the solution

L ¼ auðbzÞ: ð22Þ

When k < 0, as for an attractive Coulomb potential, u is purely imaginary and the inde-pendent variable x ¼ bz is real. u also depends on a complex ‘modular’ parameter s that isdetermined by the initial conditions [10].

The identities

L ¼ r2 dhdt;

dzdt¼ k

r2ð23Þ

allow us to determine the polar angle:

dhdz¼ 1

kL: ð24Þ

By a quadrature and a differentiation of the Painleve transcendent, r and h are both foundas functions of the parameter z, determining the orbit:

rðzÞ ¼ suðzÞduðzÞ

dz

; hðzÞ ¼ h1 þak

Z z

z1

uðbzÞdz: ð25Þ

4. Examples

4.1. A decaying orbit

To plot orbits, another form of the equations is sometimes more convenient. Definey ¼ L2 and change to h as the independent variable:

dLdh¼ ks

r;

d 1r

dh¼ � L

2ks� z

L� ks

Lr2; ð26Þ

dL2

dh¼ 2ks

Lr;

d

dhLr

� �¼ � L2

2ks� z ð27Þ

to get the third order ODE:

–50 50 100 150 200 250x

–100

–50

50

100

y

Fig. 1. A decaying orbit.


d3y

dh3þ dy

dhþ 2k2sp

y¼ 0: ð28Þ

The orbit is then given by

1

rðhÞ ¼1

ksdp

ydh

ð29Þ

We can find the orbit by numerically integrating the above third order ODE.Let us look at the particular case of the attractive Coulomb potential in more detail.

Normal units can be restored by dimensional analysis. Numerical solution of (28) allowsus to plot a slowly decaying orbit (see Fig. 1).

4.2. A capture orbit

We want a solution1 for which r ¼ uu0 > 0. Note that dz

dt < 0. If rðzÞ vanishes it must be asimple zero, since near a zero dr

dz � 1. Thus r � z� z0, L � Cðz� z0Þ where C and z0 are realconstants of integration.

If a particle approaches from infinity with angular momentum L1 and radial velocity v1,

z1 ¼ v1 þ1

2L2

1: ð30Þ

1 For simplicity, we use in this section the natural units of the problem, with jkj ¼ s ¼ 1. Normal units can berestored by dimensional analysis.

–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.5

0.5

1.0

1.5

Fig. 2. A capture orbit.


Since drdz � v1r2, we get

r � 1

v1ðz� z1Þ: ð31Þ

Thus z1 is a simple zero of u0ðzÞ.In other words, to describe the capture of an incoming charged particle by an attractive

Coulomb potential, we just have to solve the Painleve II equation with the initialconditions

z1 ¼ v1 þ1

2L2

1; uðz1Þ ¼1ffiffiffiffiffiffiffi�2p L1; u0ðz1Þ ¼ 0 ð32Þ

and evolve to a point z0 < z1 at which uðz0Þ ¼ 0. The complete orbit corresponds to thefinite range z0 6 z 6 z1 in the parameter z. We plot an example, obtained by numerical cal-culations, of such a capture orbit below (Fig. 2). The velocity blows up as 1

r2 for small r.From a finite distance the particle is captured in a finite time.

Other cases can be worked out similarly. The formulation of the Painleve equation interms of the Riemann-Hilbert problem and Isomonodromy [10,12] give powerful tech-niques to study our solution. In particular, there are a pair of conserved quantities thatreplace energy and angular momentum in this dissipative but integrable system. Also,we can generalize the Rutherford formula for Coulomb scattering to include radiation.We hope to return to such a detailed analysis in a longer paper.

Acknowledgments

I thank J. Golden, S. Iyer and A. Jordan for discussions. This work was supported inpart by the Department of Energy under the contract No. DE-FG02-91ER40685.


References

[1] I. Newton, translated by I.B. Cohen, A. Whitman, The Principia: Mathematical Principles of NaturalPhilosophy, University of California Press, 1999.

[2] J.D. Jackson, Classical Electrodynamics, third ed., Wiley, 1998.[3] F. Rohrlich, Classical Charged Particles, second ed., Addison-Wesley, Reading, MA, 1990.[4] P.A.M. Dirac, Proc. Roy. Soc. Lond. A167 (1938) 148.[5] H. Spohn, Dynamics of charged particles and their radiation field. arXiv:math-ph/9908024 (unpublished);

M. Kuze, H. Spohn, SIAM J. Math. Anal. 32 (2000) 30–53.[6] L.D. Landau, E.M. Lifshitz, Classical Theory of Fields, Section 76, Butterworth-Heinemann, 1982.[7] F. Rohrlich, Am. J. Phys. 68 (2000) 1109.[8] See for example, G.A. de Parga, R. Mares, S. Dominguez, Ann. Fond. L. de Broglie 30 (2005) 283.[9] E.L. Ince, Ordinary Differential Equations, Dover, 1956.

[10] A.S. Fokas, A.R. Its, A.A. Kapaev, V. Yu. Novokshenov, Painleve Transcendents: The Riemann-HilbertApproach AMS, Providence, RI (2006).

[11] C.A. Tracy, H. Widom, Comm. Math. Phys. 159 (1994) 151–174.[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/0508186.[15] J.C. Herrera, Phys. Rev. D15 (1977) 453–456.[16] S.G. Rajeev, Ann. Phys. 322 (2007) 1541–1555; arXiv:quant-ph/0701141.

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