On Peres approach to Fradkin-Bacry-Ruegg-Souriau’s perihelion vector

Abstract

Abstract:We solve explicitly the differential system obtained by Peres for the construction of a conserved vector associated to any central potential. We then obtain a very direct access to the discontinuous behavior of the Fradkin-Bacry-Ruegg-Souriau perihelion vector.PACS (

Fradkin-Bacry-Ruegg-Souriau perihelion vector for Gorringe-Leach equations

“…They gave an explicit construction scheme for a vector with constancy properties associated to every central potential. Nevertheless, as already noted by Bacry, Ruegg and Souriau such a vector is exceptionally one-valued, as in the Kepler case, and corresponds generally to a piecewise conserved quantity, the Fradkin-Bacry-Ruegg-Souriau (FBRS) perihelion vector (Holas and March 1990;Grandati et al 2009b).…”

“…the orbits are ellipses having a focus at O and the FBRS vector is identical with the pseudo Laplace-Runge-Lenz vector (Grandati et al 2009b):…”

“…To every radial pseudomotion is associated a piecewise conserved vector, called the Fradkin-Bacry-Ruegg-Souriau's (FBRS) vector, which is given by Grandati et al (2009b):…”

Fradkin-Bacry-Ruegg-Souriau vector in kappa-deformed space-time

“…Shortly afterwards, it was shown that this generalized conserved vector is multi-valued [6,7]. Bacry, Ruegg and Souriau showed that such a vector is exceptionally one-valued, in the Kepler case, and corresponds generally to a piecewise conserved quantity, the Fradkin-Bacry-Ruegg-Souriau (FBRS) perihelion vector [8,9]. An extension of similar construction on curved manifolds was taken up in [10].…”

“…Further, by analyzing these differential equations, it was shown that the conserved vector is multi-valued. This construction was re-visited in [9] for the central potentials in 2-dimensional space, using complex coordinates. The correspondence between Fradkin's and Peres approaches have been studied by Yoshida [12].…”