Reduction and integrability: a geometric perspective

TL;DR

This paper presents a modern geometric framework for understanding integrability and reduction in dynamical systems, emphasizing symmetries, invariant tensors, and volume forms, with applications to Hamilton-Jacobi theory.

Contribution

It develops a comprehensive geometric approach to integrability and reduction, incorporating invariant tensor fields, volume forms, and symmetry theories like Hojman, extending classical methods.

Findings

Invariance of volume forms is linked to integrability.

Hojman symmetry theory complements Noether's theorem.

Geometric perspective unifies reduction and integrability concepts.

Abstract

A geometric approach to integrability and reduction of dynamical system is developed from a modern perspective. The main ingredients in such analysis are the infinitesimal symmetries and the tensor fields that are invariant under the given dynamics. Particular emphasis is given to the existence of invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to Noether theorem and non-Noether constants of motion. The geometric approach to Hamilton-Jacobi equation is shown to be a particular example of the search for related field in a lower dimensional manifold.