Covariant Hamilton–Jacobi Formulation of Electrodynamics via Polysymplectic Reduction and Its Relation to the Canonical Hamilton–Jacobi Theory

Abstract

The covariant Hamilton–Jacobi formulation of electrodynamics is systematically derived from the first-order (Palatini-like) Lagrangian. This derivation utilizes the De Donder–Weyl covariant Hamiltonian formalism with constraints incroporating generalized Dirac brackets of forms and the associated polysymplectic reduction, which ensure manifest covariance and consistency with the field dynamics. It is also demonstrated that the canonical Hamilton–Jacobi equation in variational derivatives and the Gauss law constraint are derived from the covariant De Donder–Weyl Hamilton–Jacobi formulation after space + time decomposition.

Keywords

• covariant Hamilton–Jacobi
• Maxwell equations
• De Donder–Weyl Hamiltonian formalism
• Dirac brackets
• polysymplectic structure
• canonical Hamilton–Jacobi