Reduction and reconstruction of multisymplectic Lie systems

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 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 occurring in physics, mathematics, and control theory.

Contact Lie systems: theory and applications

“…This is more general than some other reductions appearing in the literature [30]. As far as we know, types of Marsden-Weinstein reductions have only been applied to Lie systems in [54] for multisymplectic Lie systems.…”

“…Nowadays, this has originated the study of Lie systems with VG Lie algebras of Hamiltonian vector fields relative to different types of geometric structures and their related problems. This also led to new theoretical results in differential geometry [35,54]. In particular, [6,7,18,19,21,39] analyse Lie-Hamilton systems (see [18] for Lie-Hamilton systems relative to a symplectic form).…”

“…These Poisson algebras enable us to study of integrable systems and superposition rules more efficiently than previous methods. Meanwhile, multisymplectic Lie systems, along with a particular type of multisymplectic reduction, were studied in [52,54]. Relevantly, the development of a general multisymplectic reduction has been an open problem for several decades now [11,37].…”

“…Relevantly, the development of a general multisymplectic reduction has been an open problem for several decades now [11,37]. The works [35,54] show how Lie systems lead to interesting advances in pure differential geometry too.…”

“…As a particular case, this work analyses the hereafter called contact Lie systems of Liouville type, namely contact Lie systems that are invariant relative to the Reeb vector field of their associated contact manifolds. For these systems, we introduce certain Liouville theorems and Gromov non-squeezing theorems, whose application can be considered as pioneering in the literature of Lie systems (see [34,54]). Moreover, it is remarkable that the literature on contact Hamiltonian systems is mostly focused on dissipative systems [14,15,24,26,29,41].…”