Noether symmetry groups, locally conserved integrals, and dynamical symmetries in classical mechanics

TL;DR

This paper explores the relationship between symmetries and conserved quantities in classical mechanics using a hybrid Lagrangian-Hamiltonian approach, demonstrating local integrability and explicit solutions for several systems.

Contribution

It introduces a local generalization of Liouville integrability linking symmetries to conserved integrals across different classical systems.

Findings

Local conserved integrals commute under Poisson brackets.

Action-angle variables enable explicit local integration of equations.

Symmetries lead to locally conserved quantities in diverse systems.

Abstract

Several aspects of the connection between conserved integrals (invariants) and symmetries are illustrated within a hybrid Lagrangian-Hamiltonian framework for dynamical systems. Three examples are considered: a nonlinear oscillator with time-dependent frequency (one degree of freedom); geodesics of a spheroid (two degrees of freedom); Calogero-Moser-Sutherland system of interacting particles (three degrees of freedom). For each system, a local generalization of Liouville integrability is shown. Specifically, the variational point symmetries in a Lagrangian setting lead to corresponding locally conserved integrals which are found to commute in the Poisson bracket imported from the equivalent Hamiltonian setting. Action-angle variables are then introduced in the Lagrangian setting, which leads to explicit integration of the Euler-Lagrange equations of motion locally in time.