Hamilton-Jacobi theory, Symmetries and Coisotropic Reduction

TL;DR

This paper explores the integration of reduction theory and Hamilton-Jacobi theory for Hamiltonian systems, providing a unified framework that generalizes existing methods and offers new insights into symmetries and reduction procedures.

Contribution

It introduces a reduction and reconstruction method for the Hamilton-Jacobi equation with symmetries, generalizing classical reduction procedures and unifying various solution ansatzes.

Findings

Developed a reduction and reconstruction procedure for Hamilton-Jacobi equations with symmetries.

Generalized the Ge-Marsden reduction method.

Showed classical solution ansatzes are special cases of the new framework.

Abstract

Reduction theory has played a major role in the study of Hamiltonian systems. On the other hand, the Hamilton-Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators rely on approximations of a complete solution of the Hamilton-Jacobi equation. The natural question that we address in this paper is how these two topics (reduction and Hamilton-Jacobi theory) fit together. We obtain a reduction and reconstruction procedure for the Hamilton-Jacobi equation with symmetries, even in a generalized sense to be clarified below. Several applications and relations to other reductions of the Hamilton-Jacobi theory are shown in the last section of the paper. It is remarkable that as a by-product we obtain a generalization of the Ge-Marsden reduction procedure. Quite surprinsingly, the classical ansatzs available in the literature to solve the Hamilton-Jacobi equation are also particular instances of our framework.