k -contact Lie systems: theory and applications

This paper introduces a new class of Lie systems that are Hamiltonian relative to a k-contact manifold. We show that a recent distributional approach to k-contact manifolds along with a related k-contact Hamiltonian vector field notion allow us to understand relevant Lie systems as Hamiltonian relative to a k-contact manifold. Our procedure is more general than previously known methods with this aim. As a result, we find that a plethora of Lie systems related to control and physical problems can be considered in a natural manner as k-contact Lie systems. We study their t-dependent and t-independent constants of motion, master symmetries of higher order, and other properties of interest. Finally, we use our new techniques and findings to study PDE Lie systems with a compatible k-contact manifold, some of which become Hamilton-De Donder-Weyl equations.