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Practical Introduction to Action-Dependent Field Theories

Manuel de León, Jordi Gaset Rifà, Miguel C. Muñoz-Lecanda, Xavier Rivas, Narciso Román-Roy

TL;DR

This work introduces action-dependent field theories, where the Lagrangian/Hamiltonian depend on action-encoding variables and dissipative structure, and develops a rigorous multicontact geometric framework to formulate their dynamics. It derives both Lagrangian HEL (Herglotz–Euler–Lagrange) and covariant HDW (Herglotz–Hamilton–De Donder–Weyl) equations, including a dissipation form that encodes non-conservation and generalizes Noether-type relations. The manuscript provides a comprehensive treatment of regular and singular cases via premulticontact geometry, and applies the framework to quadratic, affine Lagrangians and to diverse theories such as the wave/telegrapher equations, Klein–Gordon, Maxwell electromagnetism, metric-affine gravity, Burgers’ equation, Bosonic string, and (2+1)-dimensional gravity with Chern–Simons terms. The results highlight how action-dependent terms introduce new dynamical features and potential open-system behavior, while enabling a covariant, geometric understanding of dissipative field theories with clear avenues for constraint analysis and quantization implications.

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.

Practical Introduction to Action-Dependent Field Theories

TL;DR

This work introduces action-dependent field theories, where the Lagrangian/Hamiltonian depend on action-encoding variables and dissipative structure, and develops a rigorous multicontact geometric framework to formulate their dynamics. It derives both Lagrangian HEL (Herglotz–Euler–Lagrange) and covariant HDW (Herglotz–Hamilton–De Donder–Weyl) equations, including a dissipation form that encodes non-conservation and generalizes Noether-type relations. The manuscript provides a comprehensive treatment of regular and singular cases via premulticontact geometry, and applies the framework to quadratic, affine Lagrangians and to diverse theories such as the wave/telegrapher equations, Klein–Gordon, Maxwell electromagnetism, metric-affine gravity, Burgers’ equation, Bosonic string, and (2+1)-dimensional gravity with Chern–Simons terms. The results highlight how action-dependent terms introduce new dynamical features and potential open-system behavior, while enabling a covariant, geometric understanding of dissipative field theories with clear avenues for constraint analysis and quantization implications.

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.
Paper Structure (45 sections, 3 theorems, 194 equations)

This paper contains 45 sections, 3 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Action-dependent field theories
  3. General aspects
  4. Lagrangian formalism
  5. Hamiltonian formalism
  6. Properties of action-dependent field theories
  7. Non-conservation and symmetries.
  8. New terms in the equations.
  9. Lagrangian and Hamiltonian formalisms for action-dependent field theories
  10. Multivector fields
  11. Multicontact Lagrangian formalism
  12. Multicontact Hamiltonian formalism: The (hyper)regular case
  13. Multicontact Hamiltonian formalism: The singular case
  14. Application to general Lagrangians
  15. Quadratic (regular and singular) Lagrangians
  16. ...and 30 more sections

Key Result

Theorem 3.2

Let $({\cal P},\Theta_\mathcal{L},\omega)$ be a multicontact Lagrangian system. If ${\bf X}_\mathcal{L}$ is a semi-holonomic and integrable solution to the equations fieldLcontact (that is, a sopde), its integral sections are the solutions to the multicontact Euler--Lagrange field equations for hol

Theorems & Definitions (9)

  • Remark 3.1
  • Theorem 3.2
  • Remark 5.1
  • Remark 5.2
  • Definition B.1
  • Theorem B.2
  • Proposition B.3
  • Definition B.4
  • Definition B.5