Reduction and reconstruction of multisymplectic Lie systems

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 forms 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 from physics, mathematics, and control theory.The authors acknowledge fruitful discussions and comments from our colleague MiguelC Muñoz-Lecanda. J de Lucas acknowledges partial financial support from the Project MINIATURA 5 of the Polish National Science Centre (NCN) under Grant Number Nr 2021/05/X/ST1/01797. X Gr`acia, X Rivas, N Román-Roy, and S Vilariño acknowledge partial financial support from the Spanish Ministerio de Ciencia e Innovaci´on and Gobierno de Arag´on projectsPGC2018-098265-B-C31, PGC2018-098265-B-C33, RED2018-102541T, and E48_20R. We also thank the anonymous referees, whose interesting comments and suggestions have helped us improve our article.Peer ReviewedPostprint (published version