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Marsden--Meyer--Weinstein reduction for $k$-contact field theories

J. de Lucas, X. Rivas, S. Vilarino, B. M. Zawora

TL;DR

This work develops a Marsden--Meyer--Weinstein reduction framework for $k$-contact field theories by embedding the $k$-contact setting into an exact $k$-symplectic extension and applying a carefully adapted submanifold reduction. It defines $k$-contact momentum maps, establishes reduction-by-submanifold theorems, and proves that reduced dynamics descend to the quotient while preserving the $k$-contact structure. The authors also provide a detailed comparison with prior contact reductions, identifying and correcting key inaccuracies (notably in the choice of reduction group and tangent-space orthogonality). The resulting theory broadens the toolkit for non-conservative Hamiltonian field theories and offers concrete examples illustrating the descent of $k$-contact dynamics and the interplay between symmetry, momentum maps, and reduced geometry.

Abstract

This work devises a Marsden--Meyer--Weinstein $k$-contact reduction. Our techniques are illustrated with several examples of mathematical and physical relevance. As a byproduct, we review the previous contact reduction literature so as to clarify and to solve some inaccuracies.

Marsden--Meyer--Weinstein reduction for $k$-contact field theories

TL;DR

This work develops a Marsden--Meyer--Weinstein reduction framework for -contact field theories by embedding the -contact setting into an exact -symplectic extension and applying a carefully adapted submanifold reduction. It defines -contact momentum maps, establishes reduction-by-submanifold theorems, and proves that reduced dynamics descend to the quotient while preserving the -contact structure. The authors also provide a detailed comparison with prior contact reductions, identifying and correcting key inaccuracies (notably in the choice of reduction group and tangent-space orthogonality). The resulting theory broadens the toolkit for non-conservative Hamiltonian field theories and offers concrete examples illustrating the descent of -contact dynamics and the interplay between symmetry, momentum maps, and reduced geometry.

Abstract

This work devises a Marsden--Meyer--Weinstein -contact reduction. Our techniques are illustrated with several examples of mathematical and physical relevance. As a byproduct, we review the previous contact reduction literature so as to clarify and to solve some inaccuracies.
Paper Structure (20 sections, 37 theorems, 216 equations, 3 figures)

This paper contains 20 sections, 37 theorems, 216 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Geometry of -symplectic and -contact manifolds
  3. --vector fields and integral sections
  4. -Symplectic geometry
  5. --Symplectic Hamiltonian systems
  6. --Contact geometry
  7. --Contact Hamiltonian systems
  8. Reduction of exact --symplectic manifolds
  9. -Symplectic momentum maps
  10. -Symplectic reduction by a submanifold
  11. Marsden--Meyer--Weinstein reduction theorem for exact -symplectic manifolds
  12. Marsden--Meyer--Weinstein reduction for -contact manifolds
  13. --Contact momentum maps
  14. --Contact reduction by a submanifold
  15. --Contact Marsden--Meyer--Weinstein reduction theorem
  16. ...and 5 more sections

Key Result

Theorem 2.6

($k$-symplectic Darboux theorem) Let $(P,\bm \omega, \cal V)$ be a polarised $k$-symplectic manifold. There exists a local coordinate system around each $x\in P$, given by $\{x^i,y^\alpha_i\}$ with $i=1,\ldots, n$ and $\alpha=1,\ldots,k$, such that

Figures (3)

  • Figure 1: Diagram illustrating part of the processes to accomplish a $k$-symplectic MMW reduction. Note that $\omega_{{\bf J}^{\Phi}_{\bm\theta\alpha}}(p):=(j^\alpha_p)^*\omega^\alpha(p)$ and $\widetilde{\omega_{{\bf J}^{\Phi}_{\bm\Theta\alpha}}(p)}=\widetilde{(j^\alpha_p)^*}\widetilde{\omega^\alpha(p)}$, while $\widetilde{\omega_{{\bf J}^{\Phi}_{\bm\theta\alpha}}(p)}=\widetilde{{\rm pr}^\alpha}^*\omega_{[\mu^\alpha]}(p)$ and $\omega_{{\bf J}^{\Phi}_\alpha}(p)=({\rm pr}^{{\bf J}^\Phi_{\bm\Theta\alpha}})^*\widetilde{\omega_{{\bf J}^{\Phi}_{\bm\Theta\alpha}}(p)}$
  • Figure 2: Note that the geometric structures on the left are covers of the structures of the right in the sense that are of the form $\Omega=\mathrm{d} (s\mathop{\mathrm{pr}}\nolimits^*\eta)$.
  • Figure 3: The table above summarises the notation used in the subsequent section.

Theorems & Definitions (84)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Example 2.7: Canonical model for polarised $k$-symplectic manifolds
  • Definition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 74 more