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.

Nambu Bracket Formulation of Nonlinear Biochemical Reactions Beyond Elementary Mass Action Kinetics

“…Hydrodynamic problems have been examined [13]. Oscillatory systems such as elliptic oscillators [15,31], multi-oscillators [3,20], and other types of oscillators [4,23] have been studied in the context of Nambu mechanics. Likewise, the motion of a free particle from a Nambu perspective has been examined [4,23].…”

Oscillatory nonequilibrium Nambu systems: the canonical-dissipative Yamaleev oscillator

“…Detailed discussions on the link between the Yamaleev oscillator and the relativistic harmonic oscillator can be found in [18,19,34]. We briefly sketch here the link between the Yamaleev model (6) and the relativistic harmonic oscillator.…”

“…Then, the oscillator dynamics is given by Eq. (2), where H 1 and H 2 are Hamiltonian-like functions defined by [18,19,34] …”

Pascal Matrix Representation of Evolution of Polynomials

“…The mathematical model of composed dynamical system with finite spectrum of the energies consists of the following principal elements [14][15][16]. The state of the n-ary composed dynamical system is defined by the n-degree polynomial function f (X ).…”