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, time independent constants of motion and general properties. Our results are illustrated by 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.