Symmetries in covariant classical mechanics

Abstract

In the framework of covariant classical mechanics (i.e., general relativistic classical mechanics on a space–time with absolute time), developed by Jadczyk and Modugno, we analyze systematically the relationship between symmetries of geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure on space–time 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 by means of a covariant lift of functions of phase space. In the case of holonomic symmetries, we see that any covariant momentum map takes values in the quantizable functions in the sense of Jadczyk and Modugno, i.e., functions quadratic in velocities with leading coefficient proportional to the spacelike metric. Finally, we illustrate the results by some examples.