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

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 non-relativistic two-dimensional isotropic harmonic oscillator employing the various symplectic structures for this dynamical system recently reported.