$\omega$-Symplectic algebra and Hamiltonian vector fields

TL;DR

This paper develops a theoretical framework for $\omega$-Hamiltonian vector fields, introducing new symplectic groups and Lie algebra tools to analyze these fields in generalized coordinate systems.

Contribution

It introduces the concepts of $\omega$-symplectic and $\omega$-semisymplectic groups and explores their properties, extending classical Hamiltonian theory to more general settings.

Findings

Defined $\omega$-symplectic and $\omega$-semisymplectic groups.

Established properties of these groups and their Lie algebras.

Demonstrated applications to recognizing $\omega$-Hamiltonian vector fields.

Abstract

The purpose of this paper is presenting a theoretical basis for the study of $\omega$-Hamiltonian vector fields in a more general approach than the classical one. We introduce the concepts of $\omega$-symplectic group and $\omega$-semisymplectic group, and describe some of their properties. We show that the Lie algebra of such groups is a useful tool in the recognition of an $\omega$-Hamiltonian vector field defined on a symplectic vector space $(V,\omega)$ with respect to coordinates that are not necessarily symplectic.