Regular Reduction of Controlled Hamiltonian System with Symplectic Structure and Symmetry

TL;DR

This paper extends regular symplectic reduction theory to controlled Hamiltonian systems with symmetry, providing a unified framework for systems on cotangent bundles and reduced spaces, with applications to rigid bodies and tops.

Contribution

It introduces a new reduction framework for controlled Hamiltonian systems on symplectic fiber bundles, including regular point and orbit reductions, and explores their equivalences and applications.

Findings

Developed regular point and orbit reduction theorems for RCH systems.

Applied the theory to rigid body, heavy top, and systems with internal rotors.

Connected RCH systems to port Hamiltonian systems with symplectic structure.

Abstract

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group $\textmd{SO}(3)$ and on the Euclidean group $\textmd{SE}(3)$,as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.