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Multisymplectic structures and invariant tensors for Lie systems

X. Gràcia, J. de Lucas, M. C. Muñoz-Lecanda, S. Vilariño

TL;DR

This work introduces multisymplectic Lie systems, where a finite-dimensional Vessiot--Guldberg Lie algebra acts by Hamiltonian vector fields relative to a multisymplectic form $\Theta$. It builds a rigorous algebraic framework using tensor coalgebras, $\,\mathfrak{g}$-modules, and Casimir elements to derive invariants, constants of motion, and superposition rules for the diagonal prolongations of such systems, without solving PDEs directly. The Schwarz equation and Riccati-type diffusion/control systems serve as primary illustrations, showing how unimodular algebras yield invariant volume forms and how invariant tensor fields generate superposition rules. The approach unifies and extends previous coalgebra methods, enabling systematic construction of invariants and providing a bridge to Dirac/k-symplectic structures. Overall, the results offer a versatile toolkit for analyzing Lie systems through multisymplectic geometry with potential applications in physics, geometry, and control theory.

Abstract

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find methods to derive superposition rules, constants of motion, and invariant tensor fields relative to the evolution of the multisymplectic Lie system. Our results are illustrated with examples occurring in physics, mathematics, and control theory.

Multisymplectic structures and invariant tensors for Lie systems

TL;DR

This work introduces multisymplectic Lie systems, where a finite-dimensional Vessiot--Guldberg Lie algebra acts by Hamiltonian vector fields relative to a multisymplectic form . It builds a rigorous algebraic framework using tensor coalgebras, -modules, and Casimir elements to derive invariants, constants of motion, and superposition rules for the diagonal prolongations of such systems, without solving PDEs directly. The Schwarz equation and Riccati-type diffusion/control systems serve as primary illustrations, showing how unimodular algebras yield invariant volume forms and how invariant tensor fields generate superposition rules. The approach unifies and extends previous coalgebra methods, enabling systematic construction of invariants and providing a bridge to Dirac/k-symplectic structures. Overall, the results offer a versatile toolkit for analyzing Lie systems through multisymplectic geometry with potential applications in physics, geometry, and control theory.

Abstract

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find methods to derive superposition rules, constants of motion, and invariant tensor fields relative to the evolution of the multisymplectic Lie system. Our results are illustrated with examples occurring in physics, mathematics, and control theory.

Paper Structure

This paper contains 21 sections, 22 theorems, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Some basic concepts and notations
  3. Generalised distributions and t-dependent vector fields
  4. Lie systems
  5. Multisymplectic manifolds
  6. Unimodular Lie algebras
  7. Multisymplectic Lie systems
  8. Example: Schwarz equation
  9. Definition and main properties of multisymplectic Lie systems
  10. Example
  11. Locally automorphic Lie systems and invariant forms
  12. Locally automorphic Lie systems
  13. Invariants for locally automorphic Lie systems
  14. Example
  15. Superposition rules for multisymplectic Lie systems
  16. ...and 6 more sections

Key Result

Theorem 2.3

A $t$-dependent vector field $X$ admits a superposition rule if and only if $X$ can be written as $X = {{\sum_{\alpha=1}^r}} b_\alpha(t) X_\alpha$, for a certain family $b_1(t),\ldots,b_r(t)$ of functions and a collection $X_1,\ldots,X_r$ of vector fields spanning an $r$-dimensional real Lie algebra

Figures (2)

  • Figure 1: Diagram summarising the results of Theorem \ref{['th:SupTheo']}.
  • Figure 2: Commutative diagrams summarising the results of Theorem \ref{['th:SupTheo']} and Lemma \ref{['FuckRes']}. Both diagrams are the restrictions of diagram (\ref{['StarTrekki1']}) to the submodules of symmetric and antisymmetric tensors.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 41 more