# American Institute of Mathematical Sciences

April  2020, 13(4): 1075-1102. doi: 10.3934/dcdss.2020064

## Variational discretization of thermodynamical simple systems on Lie groups

 1 LMD - IPSL, École Normale Supérieure de Paris - PSL, 24 rue Lhomond, 75005 Paris, France 2 CNRS - LMD - IPSL, École Normale Supérieure de Paris - PSL, 24 rue Lhomond, 75005 Paris, France

* Corresponding author

Received  December 2017 Revised  August 2018 Published  April 2019

Fund Project: The authors are supported by the ANR project GEOMFLUID (ANR-14-CE23-0002).

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 [12,13], as well as its discrete counterpart whose foundations have been laid in [14]. In a first part, starting from this variational formalism on the Lie group, we perform an Euler-Poincaré 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.

Citation: Benjamin Couéraud, François Gay-Balmaz. Variational discretization of thermodynamical simple systems on Lie groups. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1075-1102. doi: 10.3934/dcdss.2020064
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##### References:
The trajectory of the center of mass of the heavy top
The different energies of the system
The relative total energy of the system. While the Runge-Kutta 2 method yields an increase in the total energy, our variational integrator displays the usual oscillatory behaviour until the system stops moving, even with a large time step
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