Contact Lie systems

TL;DR

This paper introduces contact Lie systems, exploring their properties, reductions, and applications in physics and mathematics, including Liouville theorems and non-squeezing results for systems with contact structures.

Contribution

It defines contact Lie systems, analyzes their properties, and develops theoretical tools like Liouville theorems and contact reductions, with applications to physical and mathematical examples.

Findings

Development of Liouville theorems for contact Lie systems

Application of contact reductions and Gromov non-squeezing theorems

Illustrative examples including Schwarz equations and Brockett systems

Abstract

We define and analyse the properties of contact Lie systems, namely systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of Hamiltonian vector fields relative to a contact structure. As a particular example, we study families of conservative contact Lie systems. Liouville theorems, contact reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Our results are illustrated by examples with relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, etcetera.