  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.
