Pre-multisymplectic constraint algorithm for field theories

TL;DR

This paper introduces a geometric algorithm for systematically finding consistent solutions and constraints in covariant first-order classical field theories, applicable to both Lagrangian and Hamiltonian formalisms.

Contribution

It develops an intrinsic, geometric constraint algorithm for field equations, unifying the treatment of solutions via connections or multivector fields in a pre-multisymplectic framework.

Findings

Provides a method to identify all constraint submanifolds.

Ensures solutions are integrable and holonomic where applicable.

Applies the algorithm to both Lagrangian and Hamiltonian field theories.

Abstract

We present a geometric algorithm for obtaining consistent solutions to systems of partial differential equations, mainly arising from singular covariant first-order classical field theories. This algorithm gives an intrinsic description of all the constraint submanifolds. The field equations are stated geometrically, either representing their solutions by integrable connections or, what is equivalent, by certain kinds of integrable m-vector fields. First, we consider the problem of finding connections or multivector fields solutions to the field equations in a general framework: a pre-multisymplectic fibre bundle (which will be identified with the first-order jet bundle and the multimomentum bundle when Lagrangian and Hamiltonian field theories are considered). Then, the problem is stated and solved in a linear context, and a pointwise application of the results leads to the algorithm for the general case. In a second step, the integrability of the solutions is also studied. Finally, the method is applied to Lagrangian and Hamiltonian field theories and, for the former, the problem of finding holonomic solutions is also analized.