Hamilton-Jacobi Theorems for Regular Reducible Hamiltonian Systems on a Cotangent Bundle

TL;DR

This paper develops Hamilton-Jacobi theorems for regular reducible Hamiltonian systems on cotangent bundles, extending classical results to systems with symmetry and momentum maps, with applications to Lie groups and rigid body dynamics.

Contribution

It introduces new geometric Hamilton-Jacobi theorems for reduced Hamiltonian systems, generalizing existing results to systems with symmetry and providing specific equations for Lie-Poisson systems.

Findings

Proved Hamilton-Jacobi theorems for regular reduced Hamiltonian systems.

Derived Lie-Poisson Hamilton-Jacobi equations for rigid body and heavy top.

Extended classical Hamilton-Jacobi theory to systems with symmetry and reduction.

Abstract

In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system and regular reduced Hamiltonian systems are given. At first, an important lemma is proved, and it is a modification for the corresponding result of Abraham and Marsden in [1], such that we can prove two types of geometric Hamilton-Jacobi theorem for a Hamiltonian system on the cotangent bundle of a configuration manifold, by using the symplectic structure and dynamical vector field. Then these results are generalized to the regular reducible Hamiltonian system with symmetry and momentum map, by using the reduced symplectic structure and the reduced dynamical vector field. The Hamilton-Jacobi theorems are proved and two types of Hamilton-Jacobi equations, for the regular point reduced Hamiltonian system and the regular orbit reduced Hamiltonian system, are obtained. As an application of the theoretical results, the regular point reducible Hamiltonian system on a Lie group is considered, and two types of Lie-Poisson Hamilton-Jacobi equation for the regular point reduced system are given. In particular, the Type I and Type II of Lie-Poisson Hamilton-Jacobi equations for the regular point reduced rigid body and heavy top systems are shown, respectively.