k-contact Lie systems: Theory and applications

This paper introduces a new class of Lie systems that are Hamiltonian relative to a [Formula: see text]-contact manifold. We show that a recent distributional approach to [Formula: see text]-contact manifolds along with a related [Formula: see text]-contact Hamiltonian vector field notion allows us to understand relevant Lie systems as Hamiltonian relative to a [Formula: see text]-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 [Formula: see text]-contact Lie systems. We study their [Formula: see text]-dependent and [Formula: see text]-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 [Formula: see text]-contact manifold, some of which become Hamilton–De Donder–Weyl equations.