On symmetries in covariant Galilei mechanics

In the framework of covariant classical mechanics (i.e. , generally relativistic classical mechanics on a spacetime with absolute time) developed by Jadczyk and Modugno, we analyse systematically the relations between symmetries of geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure and of its horizontal potentials are also symmetries of spacelike metric, gravitational and electromagnetic fields, Euler-Lagrange morphism and Lagrangians. Then, we provide a definition for a covariant momentum map associated with a group of cosymplectic symmetries using a covariant lift of functions of phase space. In the case when the cosymplectic symmetries projects on spacetime we see that the components of this momentum map are quantisable functions in the sense of Jadczyk and Modugno. Finally, we illustrate the results by some examples.