Multisymplectic structures and invariant tensors for Lie systems

TL;DR

This paper introduces multisymplectic structures for Lie systems, enabling new geometric methods to derive superposition rules, constants of motion, and invariant tensors, with applications across physics, mathematics, and control theory.

Contribution

It pioneers the analysis of multisymplectic Lie systems and develops geometric tools to study their invariants and solutions, expanding the theory of Lie systems.

Findings

Developed methods to derive superposition rules for multisymplectic Lie systems.

Established techniques to find constants of motion and invariant tensors.

Illustrated applications in physics, mathematics, and control theory.

Abstract

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find methods to derive superposition rules, constants of motion, and invariant tensor fields relative to the evolution of the multisymplectic Lie system. Our results are illustrated with examples occurring in physics, mathematics, and control theory.