Back to epicycles – relativistic Coulomb systems in velocity space

Abstract

Abstract. The study of relativistic Coulomb systems in velocity space is prompted by the fact that the study of Newtonian Kepler/Coulomb systems in velocity space, although less familiar than the analytic solutions in ordinary space, provides a much simpler (also more elegant) method. The simplicity and elegance of the velocity-space method derives from the linearity of the velocity equation, which is the unique feature of 1/r interactions for Newtonian and relativistic systems alike. The various types of possible trajectories are presented, their properties deduced from the orbits in velocity space, accompanied with illustrations. In particular, it is found that the orbits traversed in the relativistic velocity space (which is hyperbolic (H 3 ) rather than Euclidean) are epicyclic -circles whose centres also rotate -thus the title.PACS numbers : 03.30.+p Keywords : hodograph, relativistic Coulomb system, relativistic velocity space, Hamilton's vector, rapidity IntroductionThe motivation for the present work stems from the success of studying Newtonian Kepler/Coulomb (KC) 2-body systems in velocity space. This method was originally presented by Hamilton in 1847 [1], and elaborated years later mainly by Maxwell [2] and Feynmann [3]. It discusses the dynamics of the system by following the orbit traced by the tip of the velocity vector (termed hodograph by Hamilton). Although hardly known, the virtue of the hodograph method is that its application to classical systems with 1/r potential provides, in a very simple and elegant way, the full solution -all the necessary information regarding the dynamics of the system, including spatial trajectories -just from the discussion in velocity space. Its merits have been discusses on several occasions in the last decades [4,5,6,7,8,9,10].The success of the hodograph method with the Newtonian KC systems triggers an interest in its possible application to relativistic systems. The simplest extension of Newtonian KC systems to relativity are relativistic Coulomb systems -the limit of EM 2-body systems when one of the charges is much heavier than the other. Such systems were studied in the literature ([11, 12] and references therein), generally following standard methods in configuration space. The present work complements these studies, with a thorough discussion in relativistic velocity space.

Hamilton symmetry in relativistic Coulomb systems

“…Therefore, explicit solutions providing the hodographs and the corresponding spatial trajectories are quite immediate to get. These solutions are obtained and discussed in [14,15]. Here we proceed to present and discuss the relativistic Hamilton symmetry.…”

“…which may also be recognized as an immediate integral of ( 14) and (15). The virtue of equations ( 14), ( 15) and ( 17) is their linearity with constant coefficients in the polar representation, a unique feature of the 1/r interaction.…”

“…( 13) & ( 15)). A thorough account of the relativistic hodograph equations and solutions is contained in a recent publication ( [14]; see also [15] for an exposition of the main results).…”

The hodograph method for relativistic Coulomb systems

“…Since Ω µν = −Ω νµ , it immediately follows, as a general property of the hodograph equations, that if w µ 1 (θ) and w µ 2 (θ) are both orbits in E (1,3) that satisfy (21) then the inner product w 1 • w 2 is constant. Therefore, any solution of the hodograph equations is of constant magnitude, and their motion can only be rotational.…”

“…A brief but illustrated summary of our results is published elsewhere [21]. The full presentation splits here into two parts.…”