An overview of the Hamilton--Jacobi theory: the classical and geometrical approaches and some extensions and applications

TL;DR

This paper reviews the modern geometric formulation of Hamilton--Jacobi theory, its relation to classical approaches, and explores extensions to higher-order systems and classical field theories, highlighting its broad applicability.

Contribution

It provides a comprehensive overview of the geometric Hamilton--Jacobi theory and introduces a general framework that encompasses various physical systems and extensions.

Findings

Unified geometric description of Hamilton--Jacobi theory

Connection between modern and classical approaches clarified

Extensions to higher-order systems and field theories demonstrated

Abstract

This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton--Jacobi theory. The relation with the "classical" Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, and how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation.