Symmetries, conservation and dissipation in time-dependent contact systems

TL;DR

This paper extends Noether's theorem to non-autonomous contact Hamiltonian systems, linking symmetries with dissipated quantities and exploring their geometric structures, with applications to celestial mechanics.

Contribution

It introduces a Noether's theorem for time-dependent contact systems and characterizes symmetries related to dissipation and structure preservation.

Findings

Established a bijection between symmetries and dissipated quantities.

Identified classes of symmetries preserving contact structures.

Applied results to a two-body problem with time-dependent friction.

Abstract

In contact Hamiltonian systems, the so-called dissipated quantities are akin to conserved quantities in classical Hamiltonian systems. In this paper, we prove a Noether's theorem for non-autonomous contact Hamiltonian systems, characterizing a class of symmetries which are in bijection with dissipated quantities. We also study other classes of symmetries which preserve (up to a conformal factor) additional structures, such as the contact form or the Hamiltonian function. Furthermore, making use of the geometric structures of the extended tangent bundle, we introduce additional classes of symmetries for time-dependent contact Lagrangian systems. Our results are illustrated with several examples. In particular, we present the two-body problem with time-dependent friction, which could be interesting in celestial mechanics.