Javier de Lucas Araujo

A Lie system is a non-autonomous system of first-order ordinary differential equations whose general solution can be described as a function of a finite family of particular solutions and some constants: the so-called superposition rule. Lie systems cover as particular cases matrix Riccati equations, Smorodinsky-Winternitz oscillators, Schwarz equations, and control theory systems. The Lie--Scheffers theorem states that a Lie system amounts to a differential equation 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.

In this talk, I will introduce and analyse a class of Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. This shows that multisymplectic geometry, mainly aimed at partial differential equations and field theories, may be used to study systems of ordinary differential equations. More specifically, multisymplectic 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 geometric and coalgebra methods to derive superposition rules, constants of motion, and invariant tensor fields relative to its evolution. This extends the so-called coalgebra symmetry method for studying constants of motion of Hamiltonian systems to a much more general realm. Our results are illustrated with examples occurring in physics such as Schwarz equations or diffusion-type equations.