Reduction and reconstruction of multisymplectic Lie systems

TL;DR

This paper introduces a method using multisymplectic structures and momentum maps to reduce and reconstruct Lie systems, simplifying their analysis and solution by breaking them into more manageable subsystems, with applications across physics, mathematics, and control theory.

Contribution

It develops a novel reduction and reconstruction framework for multisymplectic Lie systems utilizing Lie symmetries and momentum maps, enabling easier analysis and solution of complex systems.

Findings

Reduction simplifies solving Lie systems.

Reconstruction retrieves original system details.

Applications demonstrated in physics, mathematics, and control theory.

Abstract

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. In this work, multisymplectic structures are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples occurring in physics, mathematics, and control theory.