k-Symplectic Lie systems: theory and applications

A Lie system is a system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot–Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the k-symplectic Lie systems, admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields with respect to the presymplectic forms of a k-symplectic structure. We devise new k-symplectic geometric methods to study their superposition rules, t-independent constants of motion and general properties. Our results are illustrated through examples of physical and mathematical interest. As a byproduct, we find a new interesting setting of application of the k-symplectic geometry: systems of first-order ordinary differential equations.