Particle dynamics on torsional galilean spacetimes

TL;DR

This paper explores particle motion on torsional galilean spacetimes, revealing a damped harmonic oscillator analogy and uncovering unique algebraic structures and geometric interpretations related to torsion effects.

Contribution

It introduces the study of torsional galilean spacetimes, analyzing their particle dynamics, symmetry algebra realization, and geometric interpretation, which were previously unexplored.

Findings

Particle motion resembles a damped harmonic oscillator influenced by torsion.

The symmetry algebra realization involves homothetic Hamiltonian vector fields and a generalized Poisson bracket.

The Bargmann extension is universal but non-central in the presence of torsion.

Abstract

We study free particle motion on homogeneous kinematical spacetimes of galilean type. The three well-known cases of Galilei and (A)dS--Galilei spacetimes are included in our analysis, but our focus will be on the previously unexplored torsional galilean spacetimes. We show how in well-chosen coordinates free particle motion becomes equivalent to the dynamics of a damped harmonic oscillator, with the damping set by the torsion. The realization of the kinematical symmetry algebra in terms of conserved charges is subtle and comes with some interesting surprises, such as a homothetic version of hamiltonian vector fields and a corresponding generalization of the Poisson bracket. We show that the Bargmann extension is universal to all galilean kinematical symmetries, but also that it is no longer central for nonzero torsion. We also present a geometric interpretation of this fact through the Eisenhart lift of the dynamics.