Hamilton-Jacobi theory for Hamiltonian systems with non-canonical symplectic structures

TL;DR

This paper develops a Hamilton-Jacobi framework for Hamiltonian systems with non-canonical symplectic structures, utilizing Dirac's method and applying it to a 2D harmonic oscillator with various symplectic forms.

Contribution

It introduces a covariant Hamilton-Jacobi approach incorporating second-class constraints and applies it to non-canonical symplectic structures in a harmonic oscillator.

Findings

Successful formulation of Hamilton-Jacobi theory with non-canonical symplectic structures.

Application to a 2D harmonic oscillator demonstrates the method's effectiveness.

Handles second-class constraints using Rothe and Scholtz's procedure.

Abstract

A proposal for the Hamilton-Jacobi theory in the context of the covariant formulation of Hamiltonian systems is done. The current approach consists in applying Dirac's method to the corresponding action which implies the inclusion of second-class constraints in the formalism which are handled using the procedure of Rothe and Scholtz recently reported. The current method is applied to the nonrelativistic two-dimensional isotropic harmonic oscillator employing the various symplectic structures for this dynamical system recently reported.