Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New Lie systems from Goursat distributions: reductions and reconstructions

Oscar Carballal

TL;DR

This work shows that certain bracket-generating Goursat distributions can produce new Lie systems when viewed as $t$-dependent vector fields. By studying the $n$-trailer system and exploiting symmetry reductions, it demonstrates a reduction to lower-dimensional Lie systems for $n=0$ (Chaplygin sleigh with $V^{X}\simeq \mathfrak{iso}(2)$) and $n=1$ (Engel case with $V^{X}\simeq \mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{iso}(2)$), aided by invariant forms that serve as principal connections. A general reconstruction theorem is developed: given a Lie system on $M$ with a $G$-invariant VG Lie algebra under a principal $G$-action, solutions of the reduced system on $M/G$ plus a Lie system on $G$ reconstruct the full dynamics via $x(t)=\Phi(g(t),\tilde{\gamma}(t))$. The framework extends Lie-system methods and superposition rules to broader ODE classes and highlights reconstruction as a practical tool for nonholonomic and geometric control problems, with Gambier and Hopf-type scenarios illustrating the approach.

Abstract

We show that types of bracket-generating distributions lead to new classes of Lie systems with compatible geometric structures. Specifically, the $n$-trailer system is analysed, showing that its associated distribution is related to a Lie system if $n = 0$ or $n = 1$. These systems allow symmetry reductions and the reconstruction of solutions of the original system from those of the reduced one. The reconstruction procedure is discussed and indicates potential extensions for studying broader classes of differential equations through Lie systems and new types of superposition rules.

New Lie systems from Goursat distributions: reductions and reconstructions

TL;DR

This work shows that certain bracket-generating Goursat distributions can produce new Lie systems when viewed as -dependent vector fields. By studying the -trailer system and exploiting symmetry reductions, it demonstrates a reduction to lower-dimensional Lie systems for (Chaplygin sleigh with ) and (Engel case with ), aided by invariant forms that serve as principal connections. A general reconstruction theorem is developed: given a Lie system on with a -invariant VG Lie algebra under a principal -action, solutions of the reduced system on plus a Lie system on reconstruct the full dynamics via . The framework extends Lie-system methods and superposition rules to broader ODE classes and highlights reconstruction as a practical tool for nonholonomic and geometric control problems, with Gambier and Hopf-type scenarios illustrating the approach.

Abstract

We show that types of bracket-generating distributions lead to new classes of Lie systems with compatible geometric structures. Specifically, the -trailer system is analysed, showing that its associated distribution is related to a Lie system if or . These systems allow symmetry reductions and the reconstruction of solutions of the original system from those of the reduced one. The reconstruction procedure is discussed and indicates potential extensions for studying broader classes of differential equations through Lie systems and new types of superposition rules.
Paper Structure (3 sections, 1 theorem, 30 equations)

This paper contains 3 sections, 1 theorem, 30 equations.

Table of Contents

  1. Introduction
  2. The $n$-trailer system
  3. Reconstruction of Lie systems

Key Result

Theorem 1

Let $X$ be a Lie system on a manifold $M$ possessing a VG Lie algebra $V$ formed by $G$-invariant vector fields with respect to a principal Lie group action $\Phi: G \times M \to M$. Let $\bm{\eta}$ be a principal connection form on the associated principal bundle $\pi: M \to M/G$. Suppose the follo Then, $\mathbb{R} \ni t \mapsto x(t):= \Phi(g(t),\widetilde{\gamma}(t))$ is a solution of $X$ such

Theorems & Definitions (2)

  • Theorem 1
  • proof