  This paper presents the continuous and discrete variational formulations of
simple thermodynamical systems whose configuration space is a (finite
dimensional) Lie group. We follow the variational approach to nonequilibrium
thermodynamics developed in \cite{GBYo2017a,GBYo2017b}, as well as its discrete
counterpart whose foundations have been laid in \cite{GBYo2017c}. In a first
part, starting from this variational formalism on the Lie group, we perform an
Euler-Poincar\'e reduction in order to obtain the reduced evolution equations
of the system on the Lie algebra of the configuration space. We obtain as
corollaries the energy balance and a Kelvin-Noether theorem. In a second part,
a compatible discretization is developed resulting in discrete evolution
equations that take place on the Lie group. Then, these discrete equations are
transported onto the Lie algebra of the configuration space with the help of a
group difference map. Finally we illustrate our framework with a heavy top
immersed in a viscous fluid modeled by a Stokes flow and proceed with a
numerical simulation.
