Multimomentum Hamiltonian Formalism

TL;DR

This paper introduces a covariant, finite-dimensional multimomentum Hamiltonian formalism for field theory, generalizing mechanics and addressing degenerate Lagrangians by automatically deriving additional conditions from Hamilton equations.

Contribution

It develops a covariant multimomentum Hamiltonian framework for field theory, extending the classical formalism to handle degenerate Lagrangians and constraints automatically.

Findings

Formalism is equivalent to Lagrangian for regular cases.

Automatically derives additional conditions for degenerate Lagrangians.

Provides a general procedure for describing constrained field systems.

Abstract

The standard Hamiltonian machinery, being applied to field theory, leads to infinite-dimensional phase spaces. It is not covariant. In this article, we present covariant finite-dimensional multimomentum Hamiltonian formalism for field theory. This is the multisymplectic generalization of the Hamiltonian formalism in mechanics. In field theory, multimomentum canonical variables are field functions and momenta corresponding to derivatives of fields with respect all world coordinates, not only the time. In case of regular Lagrangian densities, the multimomentum Hamiltonian formalism is equivalent to the Lagrangian formalism, otherwise for degenerate Lagrangian densities. In this case, the Euler-Lagrange equations become undetermined and require additional conditions which remain elusive. In the framework of the multimomentum Hamiltonian machinery, one obtaines them automatically as a part of Hamilton equations. The key point consists in the fact that, given a degenerate Lagrangian density, one must consider a family of associated multimomentum Hamiltonian forms in order to exaust solutions of the Euler-Lagrange equations. We spell out degenerate quadratic and affine Lagrangian densities. The most of field models are of these types. As a result, we get the general procedure of describing constraint field systems.