Mapping between the dynamic and mechanical properties of the relativistic oscillator and Euler free rigid body

Abstract

In this work we investigate a formal mapping between the dynamical properties of the unidimensional relativistic oscillator and the asymmetrical rigid top at a classical level. We study the relativistic oscillator within Yamaleev’s interpretation of Nambu mechanics. Such interpretation is based on the factorisation of the momenta, and as a consequence of this factorisation we are led to a three dimensional phase space. Solutions of the relativistic oscillator are given in terms of the Jacobian elliptic functions and hence we establish a correspondence of these solutions in terms of well known quantities from the rigid body theory. We also study some mechanical restrictions that appear in the mathematical development of the mapping. In particular, we find a lower bound for the relativistic frequency in order to make the mapping self-consistent and physically legitimate.