A Study of Generalized Covariant Hamilton Systems On Generalized Poisson manifold

TL;DR

This paper extends the theoretical framework of generalized Hamilton systems on manifolds by introducing a generalized gradient operator, a structural Poisson bracket, and a covariant extension, broadening their applicability.

Contribution

It proposes a generalized covariant Hamilton system incorporating a new structural Poisson bracket and a covariant extension, enhancing the theoretical completeness of Hamiltonian systems.

Findings

Defined a generalized structural Poisson bracket on manifolds.

Introduced a covariant extension form of the generalized Hamilton system.

Unified generalized Hamiltonian and S-dynamic systems under a covariant framework.

Abstract

Since the basic theoretical framework of generalized Hamilton system is not perfect and complete, there are often some practical problems that can not be expressed by generalized Hamilton system. The generalized gradient operator is defined by the structure function on manifold to improve the basic theoretical framework of the whole generalized Hamilton system. The generalized structural Poisson bracket \[\left\{ f,g \right\}={{\left\{ f,g \right\}}_{GPB}}+G\left( s,f,g \right)={{\left\{ f,g \right\}}_{GPB}}+f{{\left\{ s ,g \right\}}_{GPB}}-g{{\left\{ s ,f \right\}}_{GPB}} \]is defined as well on manifolds. The geometric bracket is also given, and the covariant extension form of the generalized Hamilton system directly related to the structure function, the generalized covariance, is further obtained--generalized covariant Hamilton system, It includes thorough generalized Hamiltonian system and S-dynamic system.