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Brackets in multicontact geometry and multisymplectization

Manuel de León, Rubén Izquierdo-López, Xavier Rivas

TL;DR

This work extends the geometry of brackets to multicontact manifolds by introducing a graded Jacobi bracket on conformal Hamiltonian forms, accompanied by two Leibniz rules and a cup product, and situates these inside a generalized pre-multisymplectization that links to multisymplectic brackets. It develops a canonical multisymplectization framework for multicontact structures, including lifting of conformal transformations, and establishes an $L_$-type relationship between the multicontact and multisymplectic bracket formalisms via a mapping $oldsymbol{ extbackslashPsi}$. The authors then formulate action-dependent (dissipative) field equations on multicontact manifolds, define good Hamiltonians through a distortion tensor, and introduce dissipated forms with evolution laws tied to a dissipation form, relating these dynamics to their multisymplectic counterparts through lifting results. These constructions are then applied to dissipative field theories, yielding explicit multicontact HDW equations in an extended phase space and showing how dissipation manifests in the evolution of observables. The work provides a foundational framework for both the geometric theory of dissipative field dynamics and potential reductions, higher Jacobi structures, and links to $k$-contact-type generalizations.

Abstract

In this paper we introduce a graded bracket of forms on multicontact manifolds. This bracket satisfies a graded Jacobi identity as well as two different versions of the Leibniz rule, one of them being a weak Leibniz rule, extending the well-known notions in contact geometry. In addition, we develop the multisymplectization of multicontact structures to relate these brackets to the ones present in multisymplectic geometry and obtain the field equations in an abstract context. The Jacobi bracket also permits to study the evolution of observables and study the dissipation phenomena, which we also address. Finally, we apply the results to classical dissipative field theories.

Brackets in multicontact geometry and multisymplectization

TL;DR

This work extends the geometry of brackets to multicontact manifolds by introducing a graded Jacobi bracket on conformal Hamiltonian forms, accompanied by two Leibniz rules and a cup product, and situates these inside a generalized pre-multisymplectization that links to multisymplectic brackets. It develops a canonical multisymplectization framework for multicontact structures, including lifting of conformal transformations, and establishes an -type relationship between the multicontact and multisymplectic bracket formalisms via a mapping . The authors then formulate action-dependent (dissipative) field equations on multicontact manifolds, define good Hamiltonians through a distortion tensor, and introduce dissipated forms with evolution laws tied to a dissipation form, relating these dynamics to their multisymplectic counterparts through lifting results. These constructions are then applied to dissipative field theories, yielding explicit multicontact HDW equations in an extended phase space and showing how dissipation manifests in the evolution of observables. The work provides a foundational framework for both the geometric theory of dissipative field dynamics and potential reductions, higher Jacobi structures, and links to -contact-type generalizations.

Abstract

In this paper we introduce a graded bracket of forms on multicontact manifolds. This bracket satisfies a graded Jacobi identity as well as two different versions of the Leibniz rule, one of them being a weak Leibniz rule, extending the well-known notions in contact geometry. In addition, we develop the multisymplectization of multicontact structures to relate these brackets to the ones present in multisymplectic geometry and obtain the field equations in an abstract context. The Jacobi bracket also permits to study the evolution of observables and study the dissipation phenomena, which we also address. Finally, we apply the results to classical dissipative field theories.
Paper Structure (12 sections, 41 theorems, 177 equations, 2 tables)

This paper contains 12 sections, 41 theorems, 177 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Brackets induced by a differential -form
  3. Multicontact structures and multisymplectization
  4. General (pre-)multisymplectizations
  5. Multisymplectizations of multicontact manifolds
  6. The mapping
  7. Multicontact field equations
  8. General multicontact Hamilton--de Donder--Weyl equations and variational multicontact manifolds
  9. Dissipated forms
  10. Multisymplectization of the dynamics
  11. Dissipative field theories
  12. Conclusions and further work

Key Result

Theorem 2.4

Let $\alpha \in \Omega^{n-p}(M)$ and $\beta \in \Omega^{n-q}(M)$ be conformal Hamiltonian forms and let $X_\alpha, V_\alpha, X_\beta$ and $V_\beta$ be multivector fields satisfying and Then, the expression only depends on $\alpha$ and $\beta$ and is also a conformal Hamiltonian form.

Theorems & Definitions (127)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Corollary 2.8
  • Definition 2.9
  • ...and 117 more