On symmetries in Galilei classical mechanics

TL;DR

This paper systematically explores the symmetries in Galilei classical mechanics, showing their relation to geometric structures and deriving a covariant momentum map, with practical examples illustrating the theoretical findings.

Contribution

It establishes the connection between symmetries of geometric objects and physical fields in Galilei mechanics and constructs a covariant momentum map for these symmetries.

Findings

Symmetries of cosymplectic structures also preserve physical fields.

A covariant momentum map is derived for symmetry groups.

Examples demonstrate the application of theoretical results.

Abstract

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