$k$-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.