Maxwell's Optics Symplectic Hamiltonian

TL;DR

This paper proposes a symplectic Hamiltonian formalism for classical field theories, addressing difficulties in applying Hamiltonian methods to gauge-invariant fields and setting the stage for future work in transformation optics.

Contribution

It introduces a reformulation of Hamiltonian formalism for source-free fields, enabling the use of symplectic methods in gauge-invariant field theories.

Findings

Reformulation allows symplectic Hamiltonian formalism for gauge-invariant fields.

Addresses difficulties in Dirac formalism with constraints.

Lays groundwork for applications in transformation optics.

Abstract

The Hamiltonian formalism is extremely elegant and convenient to mechanics problems. However, its application to the classical field theories is a difficult task. In fact, you can set one to one correspondence between the Lagrangian and Hamiltonian in the case of hyperregular Lagrangian. It is impossible to do the same in gauge-invariant field theories. In the case of irregular Lagrangian the Dirac Hamiltonian formalism with constraints is usually used, and this leads to a number of certain difficulties. The paper proposes a reformulation of the problem to the case of a field without sources. This allows to use a symplectic Hamiltonian formalism. The proposed formalism will be used by the authors in the future to justify the methods of vector bundles (Hamiltonian bundles) in transformation optics.