Successive approximations to the. singular equations. J.M. Aguirregabiria, A. Hernandez and M. Rivas

Transcription

1 Successive approximations to the regular order reduction of singular equations J.M. Aguirregabiria, A. Hernandez and M. Rivas Dept. of Theoretical Physics, The University of the Basque Country, 488 Bilbao, Spain Abstract We discuss the use of successive approximations to construct numerical approximations to the regular order reduction of singular dierential equations, such as the evolution equations appearing in classical electrodynamics and gravitation. The interest of a suitable implicit method is also suggested. I. INTRODUCTION In dierent physical theories including classical electrodynamics with radiation reaction, theories of gravitation with quadratic Lagrangian [1] and the study of quantum corrections to Einstein equations [2] there appear singular evolution equations that share some common properties. To start with, their order is higher than expected on physical grounds. For instance, theequation of motion of a charged particle with radiation reaction, the so-called Lorentz{Dirac equation, is of third order, so that initial position and velocity would not be enough to determine its evolution. As a consequence of this higher freedom when specifying the initial conditions, most of their solutions are unphysical. A natural approach to this kind of problems is provided by the concept of \order reduction", which isanevolution equation with the right order and that contains precisely the physical solutions. Order reductions have been used to replace the hereditary equations that appear in the electrodynamics of two ormore point charges [3,4] and in non-linear optics [5]. 1

2 In the context of the Lorentz{Dirac equation this concept was partially contained inlandau and Lifshitz book [6] and was clearly discussed by Kerner [7] and Sanz [8]. Several years ago, Ll.Bel wrote a routine to nd the regular order reduction of hereditary equations. To compute numerically the order reduction of singular ordinary dierential equations we have proposed and analysed in some cases a method ofsuccessive approximations [9,1]. As we will see, this last method may also deal with hereditary equations. The goal of this work is to present some of the physical motivations of our approach by means of simple but illustrative examples and toapply the numerical code we developped [11] to automatically construct the order reduction in some cases. We also point out that developing an implicit method could be of practical interest. II. HEREDITARY EQUATIONS AND SPONTANEOUS PREDICTIVISATION Ll. Bel[4] considered as simplied model of thefunctional-dierential equations of motion appearing in classical electrodynamics the hereditary equation _x t = H x t1; (1) where the subindex indicates the time at which the quantity x is evaluated. This equation is singular because when H! the number of degrees of freedom goes from 1 to 1. Bel checked numerically that the solutions of (1) become, as t! 1, (approximate) solution of _x = x ; (2) t t where is the real root (for H>1=e) of = He that is given by the principal branch of Lambert W function [12] = W (H): (3) of (1). Since all solutions of equation (2) satisfy (1) one say that (2) is an \order reduction" 2

3 Let us assume that we want to solve equation (1) by successive approximations. We start by neglecting the delay, H(tt ) _x t = Hx; t x t = xe ; (4) and using this to compute the delay in (1): H(tt 1) H He H (tt ) t t t _x = x e = He x ; x = x e : (5) By repeating this once and again, the n-th approximation is for 1 e < H<e. Hk n1 _x = Hk x ; k = e ; t n t n # as n! 1 # : : H : ( e ) eh H ( ) W (H) _x t = x t; k = e = ; H We see that the method in fact constructs the regular order reduction, which is spontaneously approached by the generic solution of the hereditary equation. III. LORENTZ{DIRAC-LIKE EQUATIONS In the same article Ll. Bel [4] considered as simplied model of the Lorentz{Dirac equation corresponding to an external force per unit mass. A more interesting (although still trivial) example may be obtained by considering the non-relativistic approximation to the Lorentz{Dirac equation (the so-called Abraham{ Lorentz equation): v _ = W + v: (6) In one dimension and with a force per unit mass given by W = a vwith a >, this reduces to _v = a v+ v: (7) 3

4 By using the method of successive approximations we can construct this regular order reduction by rst neglecting the radiation reaction, i.e. the terms proportional to, and then using each time the _v computed in the previous approximation to estimate v, obtaining _v = (v) a v = av; (8) for a < 3=4, because the recurrence 2 n+1 = 1 n, with a, is just the \logistic map", which although showing a complex behaviour, including chaos, for other parameter p values converges [13] to the xed point = =2 for 1=4 < < 3=4 and p jj < =2, and we have =1. Although in linear examples it is easy to compute explicitly the analytic expressions of the successive approximations, in more general cases they become quickly too involved and a numerical code is necessary. IV. THE NUMERICAL METHOD We have shown elsewhere [9,11] that a numerical code can be written to construct automatically the numerical successive approximations to a large class of singular equations. In particular this numerical method allows solving forward Lorentz{Dirac-like equations, even in dicult (chaotic) cases, and to construct the regular order reduction of hereditary equations. To see our routine at work, we will consider an example of interest in astrophysics [14]: an electron moving in an external magnetic eld, which for simplicity we will assume tobe uniform and constant along the electron orbit. If we choose as unit length the radius of the cyclotron orbit one obtains by neglecting the radiation reaction and the unit time is the 1 inverse of the cyclotron frequency! as = mc=eb, the Lorentz{Dirac equation can be written ( " # ) 2 (u 1 u) _ 2 2 u _ = k 2 u + u u_ u ; (9) 2 + u 1 u 4

5 2 by using the unit vector k along the magnetic eld, the parameter = E=mc that measures the initial energy, the dimensionless quantity =! and the spatial part of the four-velocity 2 2 1=2 u = (1 v=c) v=c. 6 5 In the case analysed by Shen B = 1 G, E = 1 GeV and = 1: The result obtained by means of / our generic numerical code is displayed in gure 1, which compares well with the orbit obtained by Shen by means of an analytical approximation developed to handle this particular case. FIG. 1. Forward solution of equation (9). It must be stressed, however, that the forward integration by means of our code does not suer from these numerical instabilities (which arise in the backward integration because there one has to divide by and subtract two very close quantities): in fact a very good 1 accuracy (relative error under 1 ) is obtained with a few iterations. V. IMPLICIT METHODS Let us consider a slight generalization of equation (7): _v = W(v)+ v: (1) Every order reduction _v = (v) (11) 5

6 must satisfy (v) = W (v)+ (v)(v) (12) and the successive approximation to construct the regular reduction will be n+1(v) = W (v) + n(v) n(v): (13) We can now change slightly this recurrence as follows: n+1(v) = W (v) + n(v) n+1(v); (14) because if these approximations converge their limit will be still (12). We say that a method of this kind is \implicit" because the next approximation n+1 appears implicitly in the recurrence dening it. To explore the interest of this kind of method we will use the fact that in the case of (7) the method (14) will give the same approximations for all positive values of (or ). There is another reason (apart from the domain of convergence) for which an implicit method may be better than an explicit one: even when both converge, implicit methods are often faster. Acknowledgments J.M.A. is greatly indebted to Ll. Bel for many years of teaching, hospitality and friendship. This work is a development of ideas we have learned from Prof. Bel and has proted from many discussions with him. We want to thank I. Egusquiza for having suggested the interest of implicit methods. The work of J.M.A., A.H. and M.R. has been partially supported by The University of the Basque Country under contract UPV/EHU EB36/95. J.M.A. and M.R. have also been supported by DGICYT project PB

7 REFERENCES [1] Ll. Bel and H. Sirousse-Zia, Phys. Rev. D 32, 3128 (1985). [2] L. Parker and J.Z. Simon, Phys. Rev. D 47, 1339 (1993); E. E. Flanagan and R. M. Wald, Phys. Rev. D54, 6233 (1996). [3] Ll. Bel and X. Fustero, Ann. Inst. H. Poincare 25, 411 (1976); and references therein. [4] Ll. Bel, in Relativistic Action at a Distance: Classical and Quantum Aspects, ed. J. Llosa Springer, Berlin (1982), p. 21. [5] Ll. Bel, J.-L. Boulanger and N. Deruelle, Phys. Rev. A 37, 1563 (1988). [6] L. Landau and L. Lifshitz, The Classical Theory of Fields, Addison-Wesley, New York (1951). [7] E. Kerner, J. Math. Phys (1965). [8] J.L. Sanz, J. Math. Phys. 2, 2334 (1979). [9] J.M. Aguirregabiria, J. Phys. A 3, 2391 (1997). [1] J.M. Aguirregabiria, A. Hernandez and M. Rivas, J. Phys. A 3, L651 (1997). [11] J.M. Aguirregabiria, Ll. Bel, A. Hernandez and M. Rivas, Regular order reductions of ordinary and delay-dierential equations, preprint (1998). [12] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jerey and D.E. Knuth, On the Lambert W Function, Technical Report CS-93-3, Dept. Comp. Sci., University ofwaterloo (1993). [13] E. Ott, Chaos in dynamical systems, Cambridge University Press, New York (1993). [14] C.S. Shen, Phys. Rev. Lett. 24, 41 (197); Phys. Rev. D, 6, 2736(1972); Phys. Rev. D, 17, 434 (1978). 7