Periodic first integrals for Hamiltonian systems of Lie type

TL;DR

This paper establishes the existence of Lie algebra structures of first integrals for time-dependent Hamiltonian systems of Lie type, including periodic systems, with applications to nonlinear oscillator dynamics.

Contribution

It introduces a Lie algebra framework for first integrals in time-dependent Hamiltonian systems of Lie type, extending to periodic cases using Floquet theory.

Findings

Existence of a Lie algebra of first integrals for systems of Lie type.

Periodic Hamiltonian systems have an abelian Lie algebra of periodic first integrals.

Application demonstrated on nonlinear oscillator dynamics.

Abstract

We prove the existence of a Lie algebra of first integrals for time dependent Hamiltonian systems of Lie type. Moreover, applying the Floquet theory for periodic Euler systems on Lie algebras, we show the existence of an abelian Lie algebra of periodic first integrals for periodic Hamiltonian systems. An application to the dynamics of a nonlinear oscillator is given.