A relativistic analogue of the Kepler problem

Abstract

The Poincar~ invariant system of two point particles with an instantaneous interactionat-a-distance originally proposed by Fokker is studied in the Hamittonian formalism. The interaction, which agrees to first order in the coupling constant with the electromagnetic one obtained from the Lienard-Wiechert fields, is described in an advanced-retarded state space. The first particle moves in the advanced field of the second which in turn is subject to the retarded field of the first. The acceleration terms in the Lidnard-Wiechert fields are neglected.In this theory the state space of the system is a twelve-dimensional manifold ~ and the motions are described as integral curves of a vector field that is obtained as the projection of the generator of time translations in space-time. The Poincard group acts on this manifold 2 in a well-defined way and leaves a symplectie form co invariant. Thus the set of all possible motions of this system can be studied by the methods of modern symplectic mechanics. In this paper the general method is explained and the set of all bounded motions for two equal rest masses and an attractive force is studied qualitatively and numerically, tn the limit (binding energy)/(sum of rest masses ). ( speed of light) 2 ~ 0 all the features of the classical Kepler motion are obtained. generated b y :T leaves co invariant, i.e. ~o c o = 0, w h e n c e (locally) ~f J co = dH for some f u n c t i o n H o n £ t h a t can be called the H a m l l t o n i a n . F o r an isolated relativistic system co is m o r e o v e r required to be invariant u n d e r the Poincar4 group.

Isotropic Forms of Dynamics in the Relativistic Direct Interaction Theory

“…These models are based on the Fokker-type integrals (5.1) with the Fokker potentials U = U v and U = U s , respectively (see (7.3)). Both scalar and vector models were partly considered earlier (the former in the two-dimensional space-time only) [ 44,45,46,26,20,28,47]. Our results obtained by means of both the Lagrangian and especially the Hamiltonian formulations of these models complete the analysis of their classical dynamics.…”

“…where ǫ = ±1. Such a description has been developed within the framework of the predictive relativistic mechanics in a series of papers by Künzle [ 27,28,29]. The idea of this definition of simultaneity was formulated in the classic Van Dam-Wigner's work [ 30].…”

“…These relations have been used in [ 27,28]. When we choose σ(x) as in the front form [ϕ(t, y) = t − y 3 ], we obtain…”

“…The structure of the motion integrals (4.31), (4.32) agrees with the results of Refs. [ 27,28] which were derived within the framework of the predictive relativistic dynamics.…”

Fokker-Type Confinement Models From Effective Lagrangian in Classical Yang–mills Theory

“…Rudd and Hill, 27 and studied in more detail in Ref. 28. For NT the corresponding time-asymmetric Fokker-type action can be taken as the classical base for relativistic confinement model.…”