Noether symmetries and integrability in time-dependent Hamiltonian mechanics

TL;DR

This paper explores Noether symmetries in time-dependent Hamiltonian mechanics, providing explicit formulas, examples like the Kepler problem, and a variant of the integrability theorem based on these symmetries.

Contribution

It introduces a new perspective on Noether symmetries as transformations preserving the Poincaré-Cartan form and proves a variant of the integrability theorem for time-dependent systems.

Findings

Explicit expression for symmetries in contact cases

Application to natural mechanical systems like Kepler problem

A new variant of the integrability theorem

Abstract

We consider Noether symmetries within Hamiltonian setting as transformations that preserve Poincar\'e-Cartan form, i.e., as symmetries of characteristic line bundles of nondegenerate 1-forms. In the case when the Poincar\'e-Cartan form is contact, the explicit expression for the symmetries in the inverse Noether theorem is given. As examples, we consider natural mechanical systems, in particular the Kepler problem. Finally, we prove a variant of the theorem on complete (non-commutative) integrability in terms of Noether symmetries of time-dependent Hamiltonian systems.