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$k$-contact Lie systems: theory and applications

Javier de Lucas, Xavier Rivas, Tomasz Sobczak

TL;DR

This work introduces $k$-contact Lie systems, a generalization of contact Lie systems, by formulating Lie systems as $m\eta$-Hamiltonian vector fields on co-oriented $k$-contact manifolds and exploiting a distributional approach. It develops a comprehensive framework for $t$-dependent and generalised constants of motion, master symmetries, and diagonal prolongations, and relates these systems to other geometric structures such as $k$-symplectic and presymplectic settings. The authors provide a broad array of applications, including control-theoretic models (e.g., a 5D control system, a degree-three Brockett generalisation, and a front-wheel car) and complex PDE Lie systems that admit Hamilton--De Donder--Weyl interpretations, illustrating the practical impact of the theory. They also extend the approach to PDE Lie systems, showing how $k$-contact geometry yields new Hamiltonian formulations and potential coalgebra methods for constructing superposition rules. The results offer new geometric tools for analyzing dissipative and constrained Lie systems and point toward rich extensions in PDEs and higher-dimensional $k$-contact structures.

Abstract

This paper introduces a new class of Lie systems that are Hamiltonian relative to a $k$-contact manifold. We show that a recent distributional approach to $k$-contact manifolds along with a related $k$-contact Hamiltonian vector field notion allow us to understand relevant Lie systems as Hamiltonian relative to a $k$-contact manifold. Our procedure is more general than previously known methods with this aim. As a result, we find that a plethora of Lie systems related to control and physical problems can be considered in a natural manner as $k$-contact Lie systems. We study their $t$-dependent and $t$-independent constants of motion, master symmetries of higher order, and other properties of interest. Finally, we use our new techniques and findings to study PDE Lie systems with a compatible $k$-contact manifold, some of which become Hamilton--De Donder--Weyl equations.

$k$-contact Lie systems: theory and applications

TL;DR

This work introduces -contact Lie systems, a generalization of contact Lie systems, by formulating Lie systems as -Hamiltonian vector fields on co-oriented -contact manifolds and exploiting a distributional approach. It develops a comprehensive framework for -dependent and generalised constants of motion, master symmetries, and diagonal prolongations, and relates these systems to other geometric structures such as -symplectic and presymplectic settings. The authors provide a broad array of applications, including control-theoretic models (e.g., a 5D control system, a degree-three Brockett generalisation, and a front-wheel car) and complex PDE Lie systems that admit Hamilton--De Donder--Weyl interpretations, illustrating the practical impact of the theory. They also extend the approach to PDE Lie systems, showing how -contact geometry yields new Hamiltonian formulations and potential coalgebra methods for constructing superposition rules. The results offer new geometric tools for analyzing dissipative and constrained Lie systems and point toward rich extensions in PDEs and higher-dimensional -contact structures.

Abstract

This paper introduces a new class of Lie systems that are Hamiltonian relative to a -contact manifold. We show that a recent distributional approach to -contact manifolds along with a related -contact Hamiltonian vector field notion allow us to understand relevant Lie systems as Hamiltonian relative to a -contact manifold. Our procedure is more general than previously known methods with this aim. As a result, we find that a plethora of Lie systems related to control and physical problems can be considered in a natural manner as -contact Lie systems. We study their -dependent and -independent constants of motion, master symmetries of higher order, and other properties of interest. Finally, we use our new techniques and findings to study PDE Lie systems with a compatible -contact manifold, some of which become Hamilton--De Donder--Weyl equations.

Paper Structure

This paper contains 18 sections, 23 theorems, 204 equations, 1 table.

Table of Contents

  1. Introduction
  2. Basics on Lie systems
  3. --Vector fields and its integral sections
  4. k-contact geometry
  5. -contact Lie systems
  6. Generalised t-dependent constants of the motion for k-contact Lie systems
  7. -contact Lie systems and other geometric structures
  8. Methods to construct k-contact Lie systems
  9. Diagonal prolongation of k-contact Lie systems
  10. Applications of k-contact Lie systems
  11. A control Lie system
  12. The complex Schwarz equation
  13. Generalisation to degree three of the Brockett system
  14. A new control system
  15. Front-wheel driven kinematic car
  16. ...and 3 more sections

Key Result

Theorem 2.1

A $t$-dependent system $X$ on a manifold $M$ admits a superposition rule eq:Superposition_rule if, and only if, $X$ can be written as where $X_{1}, \ldots, X_{r}$ is a family of vector fields on $M$ spanning an r-dimensional real Lie algebra of vector fields $V$, a so called Vessiot--Guldberg Lie algebra of $X$ on $M$, and $b_{1}(t),\ldots,b_{r}(t)$ are arbitrary $t$-dependent functions.

Theorems & Definitions (63)

  • Theorem 2.1: Lie Theorem
  • Definition 2.2
  • Definition 2.3
  • Definition 4.1
  • Theorem 4.2: Reeb vector fields GGMRR_20
  • Definition 4.3
  • Definition 4.4
  • Proposition 4.5
  • Definition 4.6
  • Definition 4.7
  • ...and 53 more