Multisymplectic structures and invariant tensors for Lie systems

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.

33 pages

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• Geometric models for Lie--Hamilton systems on ${\mathbb{R}}^2$
• New Lie systems from Goursat distributions: reductions and reconstructions
• Reduction and reconstruction of multisymplectic Lie systems
• An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold
• Stratified Lie systems: Theory and applications