A hybrid Lagrangian-Hamiltonian framework and its application to conserved integrals and symmetry groups

TL;DR

This paper introduces a hybrid Lagrangian-Hamiltonian framework that unifies symmetry and conservation principles, enabling analysis of integrable systems without explicit Lagrangians and clarifying symmetry types.

Contribution

It presents a modern Noether's theorem using only equations of motion, formulates the Poisson bracket with Lagrangian variables, and treats autonomous and non-autonomous systems uniformly.

Findings

Derived a form of Noether's theorem based solely on equations of motion.

Formulated the Poisson bracket using Lagrangian variables.

Applied the framework to find symmetry groups of Liouville integrable systems.

Abstract

A hybrid framework is developed that highlights and unifies the most important aspects of the Noether correspondence between symmetries and conserved integrals in Lagrangian and Hamiltonian mechanics. Several main results are shown: (1) a modern form of Noether's theorem is presented that uses only the equations of motion, with no knowledge required of an explicit Lagrangian; (2) the Poisson bracket is formulated with Lagrangian variables and used to express the action of symmetries on conserved integrals; (3) features of point symmetries versus dynamical symmetries are clarified and explained; (4) both autonomous and non-autonomous systems are treated on an equal footing. These results are applied to dynamical systems that are locally Liouville integrable. In particular, they allow finding the complete Noether symmetry group of such systems.