Particle dynamics on torsional galilean spacetimes

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 space -times. We show how in well-chosen coordinates free particle motion becomes equiva-lent to the dynamics of a damped harmonic oscillator, with the damping set by the tor-sion. 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.

Automated summary

In this article, we study the motion of a free particle on homogeneous kinematical spacetimes of Galilean type. We analyze the well-known cases of Galilei and (A)dS-Galilei spacetimes, but focus on the previously unexplored torsional Galilean spacetimes. We demonstrate that, in properly chosen coordinates, the free particle motion is equivalent to the dynamics of a damped harmonic oscillator, with the damping induced by the torsion. We also uncover the subtle realization of the kinematical symmetry algebra in terms of conserved charges and present interesting surprises in terms of homothetic versions of Hamiltonian vector fields and a generalized Poisson bracket.