q-Cosymplectic Geometry, Integrability and Reduction

In the present paper, we define the concept of a $q$-cosymplectic manifold, on which we study the Hamiltonian, gradient, local gradient, and $q$-evolution vector fields. Several Liouville--Arnold-type theorems and a $q$-cosymplectic Marsden--Weinstein reduction theorem are established. We also provide physical examples illustrating the application of the structure to multitime dynamics (Fast-slow dynamical system). To make our work more self-contained, we include detailed proofs for some results that may resemble those known for cosymplectic manifolds.