Metriplectic Euler-Poincar\'e equations: smooth and discrete dynamics

In this paper we will study some interesting properties of modifications of the Euler-Poincar\'e equations when we add a special type of dissipative force, so that the equations of motion can be described using the metriplectic formalism. The metriplectic representation of the dynamics allows us to describe the conservation of energy, as well as to guarantee entropy production. Moreover, we describe the use of discrete gradient systems to numerically simulate the evolution of the continuous metriplectic equations preserving their main properties: preservation of energy and correct entropy production rate.