Normal forms of $\omega$-Hamiltonian vector fields with symmetries

TL;DR

This paper develops algebraic methods to derive normal forms of omega-Hamiltonian vector fields that respect symmetries and reversing symmetries, combining classical Hamiltonian normal form techniques with group invariant theory.

Contribution

It introduces a novel algebraic approach to obtain symmetry-preserving normal forms of omega-Hamiltonian vector fields under semisymplectic group actions.

Findings

Normal forms preserve Hamiltonian structure.

Normal forms respect original symmetries and reversing symmetries.

Method integrates classical Hamiltonian normal forms with invariant group theory.

Abstract

In this paper, we present algebraic tools to obtain normal forms of $\omega$-Hamiltonian vector fields under a semisymplectic action of a Lie group, by taking into account the symmetries and reversing symmetries of the vector field. The normal forms resulting from the process preserve the Hamiltonian condition and the types of symmetries of the original vector field. Our techniques combine the classical method of normal forms of Hamiltonian vector fields with the invariant theory of groups.