Hamilton-Jacobi Equations for Nonholonomic Reducible Hamiltonian Systems on a Cotangent Bundle

TL;DR

This paper develops Hamilton-Jacobi theory for nonholonomic reducible Hamiltonian systems on cotangent bundles, establishing geometric conditions and equations that describe their dynamics and symmetries, with illustrative examples.

Contribution

It introduces new Hamilton-Jacobi equations for nonholonomic systems, including reduced systems with symmetries, and proves related theorems with applications.

Findings

Derived geometric constraint conditions for distributional two-forms.

Established two types of Hamilton-Jacobi equations for nonholonomic systems.

Provided examples illustrating the theoretical results.

Abstract

In this paper, for a variety of nonholonomic (reducible) Hamiltonian systems, we first give to various distributional Hamiltonian systems, by analyzing carefully the dynamics and structures of the nonholonomic Hamiltonian systems. Secondly, we derive precisely the geometric constraint conditions of the induced distributional two-form for the nonholonomic dynamical vector field, which are called the Type I and Type II of Hamilton-Jacobi equations. Thirdly, we generalize the above results for the nonholonomic reducible Hamiltonian systems with symmetries, as well as with momentum maps, and prove two types of Hamilton-Jacobi theorems for various nonholonomic reduced distributional Hamiltonian systems. Finally, as an application, we give two examples to illustrate the theoretical results. These researches reveal the deeply internal relationships of the nonholonomic constraints, the induced (resp. reduced) distributional two-forms and the dynamical vector fields of the nonholonomic Hamiltonian system and its various distributional Hamiltonian systems.