Geometric Hamilton-Jacobi Theory

TL;DR

This paper revisits the Hamilton-Jacobi theory focusing on bi-Hamiltonian structures, formulating it on tangent bundles to handle both regular and singular Lagrangian systems, with applications to classical physical models.

Contribution

It introduces a formulation of the Hamilton-Jacobi problem on tangent bundles, extending it to singular Lagrangians without secondary constraints, and explores bi-Hamiltonian structures.

Findings

Developed Hamilton-Jacobi formulation on tangent bundles.

Extended the theory to singular Lagrangian systems.

Applied the framework to classical models like the relativistic particle and rigid body.

Abstract

The Hamilton-Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed for systems described by regular Lagrangians and then extended to systems described by singular Lagrangians with no secondary constraints. We also consider the example of the free relativistic particle, the rigid body and the electron-monopole system.