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Symmetries and Noether's theorem for multicontact field theories

Xavier Rivas, Narciso Román-Roy, Bartosz M. Zawora

TL;DR

This work develops a geometric extension of Noether's theorem to action-dependent (dissipative) field theories through multicontact geometry, defining conserved and dissipated quantities and clarifying the role of generalized and Noether symmetries in both Lagrangian and Hamiltonian formalisms. It introduces the dissipation mechanism via the dissipation form $\\sigma_{\\Theta}$ and the modified derivative $\\overline{d}$, and proves a Noether-type result: for an infinitesimal Noether symmetry with $\\boldsymbol{\\xi}_Y=\\iota_Y\\Theta_{\\cal L}$, one has $\\iota_{\\mathbf{X}}\\overline{d}\\boldsymbol{\\xi}_Y=0$, linking symmetries to dissipated quantities and their laws. The framework is illustrated with a damped vibrating string, where a strong Noether symmetry yields a dissipative current that reproduces the damped wave equation, and the Hamiltonian counterpart is shown to admit analogous structures via canonical lifts. Overall, the paper generalizes Noether's theorem to multicontact field theories, bridging symmetry analysis and dissipation in both Lagrangian and Hamiltonian settings and suggesting avenues for further development, including singular theories and reductions.

Abstract

A geometric framework, called multicontact geometry, has recently been developed to study action-dependent field theories. In this work, we use this framework to analyze symmetries in action-dependent Lagrangian and Hamiltonian field theories, as well as their associated dissipation laws. Specifically, we establish the definitions of conserved and dissipated quantities, define the general symmetries of the field equations and the geometric structure, and examine their properties. The latter ones, referred to as Noether symmetries, lead to the formulation of a version of Noether's Theorem in this setting, which associates each of these symmetries with the corresponding dissipated quantity and the resulting conservation law.

Symmetries and Noether's theorem for multicontact field theories

TL;DR

This work develops a geometric extension of Noether's theorem to action-dependent (dissipative) field theories through multicontact geometry, defining conserved and dissipated quantities and clarifying the role of generalized and Noether symmetries in both Lagrangian and Hamiltonian formalisms. It introduces the dissipation mechanism via the dissipation form and the modified derivative , and proves a Noether-type result: for an infinitesimal Noether symmetry with , one has , linking symmetries to dissipated quantities and their laws. The framework is illustrated with a damped vibrating string, where a strong Noether symmetry yields a dissipative current that reproduces the damped wave equation, and the Hamiltonian counterpart is shown to admit analogous structures via canonical lifts. Overall, the paper generalizes Noether's theorem to multicontact field theories, bridging symmetry analysis and dissipation in both Lagrangian and Hamiltonian settings and suggesting avenues for further development, including singular theories and reductions.

Abstract

A geometric framework, called multicontact geometry, has recently been developed to study action-dependent field theories. In this work, we use this framework to analyze symmetries in action-dependent Lagrangian and Hamiltonian field theories, as well as their associated dissipation laws. Specifically, we establish the definitions of conserved and dissipated quantities, define the general symmetries of the field equations and the geometric structure, and examine their properties. The latter ones, referred to as Noether symmetries, lead to the formulation of a version of Noether's Theorem in this setting, which associates each of these symmetries with the corresponding dissipated quantity and the resulting conservation law.

Paper Structure

This paper contains 13 sections, 12 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review on multicontact geometry
  4. Multicontact Lagrangian systems
  5. Multicontact Hamiltonian systems (regular case)
  6. Conservation and dissipation forms and symmetries
  7. Conservation and dissipation laws
  8. Generalized symmetries
  9. Noether symmetries, Noether's theorem
  10. Conservation and dissipation laws and symmetries for multicontact Hamiltonian systems
  11. Application: a vibrating string with a linear damping
  12. Conclusions and outlook
  13. Multivector fields on manifolds and fiber bundles

Key Result

Proposition 2.3

Let $(P,\Theta,\omega)$ be a (pre)multicontact manifold, then there exists a unique $\sigma_{\Theta}\in\Omega^1(P)$, called the dissipation form, satisfying

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Remark 2.9
  • Definition 3.1
  • ...and 30 more