Skinner--Rusk formalism of action-dependent multicontact field theories
Xavier Rivas, Narciso Román-Roy, Annamaria Villanova
TL;DR
The paper addresses the need for a covariant framework to describe action-dependent (non-conservative) classical field theories. It develops multicontact geometry as a generalization of contact and multisymplectic formalisms and introduces a Skinner–Rusk–type unified approach that merges the Lagrangian and Hamiltonian descriptions within action-dependent dynamics, including a constraint algorithm for singular theories. The main contributions are the systematic development of multicontact Lagrangian and Hamiltonian formalisms, and the Skinner–Rusk unified framework implemented on extended jet–multimomentum bundles with a final constraint submanifold, illustrated by Maxwell theory with action-dependent terms describing electromagnetism in matter. This work provides a robust geometric toolkit for dissipative field theories and opens avenues for extensions to higher-order theories and gauge-field models, with potential applications in modified gravitation and electromagnetism in complex media.
Abstract
The newly developed multicontact structure, based on contact and multisymplectic geometries, provides a very general geometrical framework suitable for the treatment of action-dependent classical field theories. Having successfully applied it to formulate the Lagrangian and Hamiltonian descriptions of these theories, in the present work, the well-known Skinner--Rusk formalism is presented in this multicontact setting, which allows us to provide a combined version of both Lagrangian and Hamiltonian formalisms particularly suitable for the study and description of singular theories. As an application of this last situation, we study a modification of Maxwell's Lagrangian of classical electromagnetism, which incorporates action-dependent terms and allows us to describe electromagnetism in material media.