Practical Introduction to Action-Dependent Field Theories

TL;DR

This paper introduces action-dependent field theories, a class that models non-conservative and open systems using multicontact geometry, with detailed examples and a constraint algorithm for singular cases.

Contribution

It presents a new mathematical framework for action-dependent field theories based on multicontact geometry, expanding the modeling of non-conservative systems.

Findings

Applied the formalism to classical equations like wave, Klein-Gordon, and heat equations.

Analyzed singular and regular Lagrangians within the new framework.

Provided detailed constraint algorithms for singular cases.

Abstract

Action-dependent field theories are systems where the Lagrangian or Hamiltonian depends on new variables that encode the action. They model a larger class of field theories, including non-conservative behavior, while maintaining a well-defined notion of symmetries and a Noether theorem. This makes them especially suited for open systems. After a conceptual introduction, we make a quick presentation of a new mathematical framework for action-dependent field theory: multicontact geometry. The formalism is illustrated with a variety of action-dependent Lagrangians, some of which are regular and others singular, derived from well-known theories whose Lagrangians have been modified to incorporate action-dependent terms. Detailed computations are provided, including the constraint algorithm for the singular cases, in both the Lagrangian and Hamiltonian formalisms. These are the one-dimensional wave equation, the Klein-Gordon equation and the telegrapher equation, Maxwell's electromagnetism, Metric-affine gravity, the heat equation and Burguers' equation, the Bosonic string theory, and (2+1)-dimensional gravity and Chern-Simons equation.