On the non-Abelian Lie bracket and the generalized covariant Hamilton system

TL;DR

This paper introduces a generalized geometric Lie bracket based on non-Abelian Lie algebra to develop a structural Poisson bracket and explores second order equations in the generalized covariant Hamilton system.

Contribution

It proposes a new generalized geometric Lie bracket and discusses its application to second order equations in covariant Hamilton systems.

Findings

Introduction of a generalized geometric Lie bracket.

Development of a structural Poisson bracket.

Analysis of second order equations in covariant Hamilton systems.

Abstract

Based on the non-Abelian Lie algebra, a generalized geometric Lie bracket on vector space is proposed to further realize the generalized structural Poisson bracket, and then we briefly discuss the second order equations of the generalized covariant Hamilton system with respect to the time and coordinates and its simple application.