Particle dynamics on torsional galilean spacetimes

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