# 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 and Continuous Dynamical Systems - S, 2020, 13 (4) : 1075-1102. doi: 10.3934/dcdss.2020064
##### References:
 [1] W. Bauer and F. Gay-Balmaz, Towards a variational discretization of compressible fluids: The rotating shallow water equations, J. Comp. Dyn, accepted, https://arXiv.org/pdf/1711.10617.pdf doi: 10.3934/jcd.2019001. [2] A. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and double bracket dissipation, Com. Math. Phys., 175 (1996), 1-42.  doi: 10.1007/BF02101622. [3] A. I. Bobenko and Y. S. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products, Lett. Math. Phys., 49 (1999), 79-93.  doi: 10.1023/A:1007654605901. [4] N. Bou-Rabee, Hamilton-Pontryagin Integrators on Lie Groups, Ph.D thesis, California Institute of Technology, 2007, http://resolver.caltech.edu/CaltechETD:etd-06052007-153115. [5] N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups Part Ⅰ: Introduction and structure-preserving properties, Foundations of Computational Mathematics, 9 (2009), 197-219.  doi: 10.1007/s10208-008-9030-4. [6] H. Brenner and J. Happel, Low Reynolds Number Hydrodynamics, Mechanics of fluids and transport processes, 1, Martinus Nijhoff publishers, 1983. [7] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Memoirs of the AMS, 152 (2001), x+108 pp. doi: 10.1090/memo/0722. [8] M. Desbrun, E. Gawlik, F. Gay-Balmaz and V. Zeitlin, Variational discretization for rotating stratified fluids, Disc. Cont. Dyn. Syst. Series A, 34 (2014), 477-509.  doi: 10.3934/dcds.2014.34.477. [9] E. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D, 240 (2011), 1724-1760.  doi: 10.1016/j.physd.2011.07.011. [10] F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Adv. Appl. Math., 42 (2009), 176-275.  doi: 10.1016/j.aam.2008.06.002. [11] F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems, Physica D: Nonlinear Phenomena, 239 (2010), 1929-1947.  doi: 10.1016/j.physd.2010.07.002. [12] F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part Ⅰ: Discrete systems, J. Geom. Phys., 111 (2017), 169-193.  doi: 10.1016/j.geomphys.2016.08.018. [13] F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part Ⅱ: Continuum systems, J. Geom. Phys., 111 (2017), 194-212.  doi: 10.1016/j.geomphys.2016.08.019. [14] F. Gay-Balmaz and H. Yoshimura, Variational discretization for the nonequilibrium thermodynamics of simple systems, Nonlinearity, 31 (2018), 1673-1705.  doi: 10.1088/1361-6544/aaa10e. [15] F. Gay-Balmaz and H. Yoshimura, A variational formulation of nonequilibrium thermodynamics for discrete open systems with mass and heat transfer, Entropy, 20 (2018), Paper No. 163, 26 pp. doi: 10.3390/e20030163. [16] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31, Springer, 2006. [17] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721. [18] D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry, From Finite to Infinite Dimensions, Oxford Texts in Applied and Engineering Mathematics, 12, Oxford University Press, Oxford, 2009. [19] C. Kane, J. E. Marsden and M. Ortiz, Symplectic-energy-momentum preserving variational integrators, J. Math. Phys., 40 (1999), 3353-3371.  doi: 10.1063/1.532892. [20] S. Kim and S. Karrila, Microhydrodynamics: Principles and Selected Applications, Dover, 1991. [21] H. Lamb, Hydrodynamics, 6th revised edition, Cambridge University Press, Cambridge, 1993. [22] M. de León and D. Martín De Diego, Variational integrators and time-dependent Lagrangian systems, Rep. Math. Phys., 49 (2002), 183-192.  doi: 10.1016/S0034-4877(02)80017-9. [23] R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlin. Sci., 16 (2006), 283-328.  doi: 10.1007/s00332-005-0698-1. [24] J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.  doi: 10.1088/0951-7715/12/6/314. [25] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, A Basic Exposition of Classical Mechanical Systems, Second edition. Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. [26] J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, ZAMP, 44 (1993), 17-43.  doi: 10.1007/BF00914351. [27] J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Comm., 1 (1993), 139-164. [28] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [29] D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden and M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D: Nonlinear Phenomena, 240 (2011), 443-458.  doi: 10.1016/j.physd.2010.10.012. [30] A.-T. Petit and P.-L. Dulon, Recherches sur quelques points importants de la théorie de la chaleur, Annales de Chimie et de Physique, 10 (1819), 395-413. [31] R. W. Sharpe, Differential geometry, Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997. [32] E. C. G. Stueckelberg and P. B. Scheurer, Thermocinétique Phénoménologique Galiléenne, Birkhäuser, 1974. [33] V. Zeitlin, Finite-mode analogues of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure, Physica D, 49 (1991), 353-362.  doi: 10.1016/0167-2789(91)90152-Y.

show all references

##### References:
 [1] W. Bauer and F. Gay-Balmaz, Towards a variational discretization of compressible fluids: The rotating shallow water equations, J. Comp. Dyn, accepted, https://arXiv.org/pdf/1711.10617.pdf doi: 10.3934/jcd.2019001. [2] A. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and double bracket dissipation, Com. Math. Phys., 175 (1996), 1-42.  doi: 10.1007/BF02101622. [3] A. I. Bobenko and Y. S. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products, Lett. Math. Phys., 49 (1999), 79-93.  doi: 10.1023/A:1007654605901. [4] N. Bou-Rabee, Hamilton-Pontryagin Integrators on Lie Groups, Ph.D thesis, California Institute of Technology, 2007, http://resolver.caltech.edu/CaltechETD:etd-06052007-153115. [5] N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups Part Ⅰ: Introduction and structure-preserving properties, Foundations of Computational Mathematics, 9 (2009), 197-219.  doi: 10.1007/s10208-008-9030-4. [6] H. Brenner and J. Happel, Low Reynolds Number Hydrodynamics, Mechanics of fluids and transport processes, 1, Martinus Nijhoff publishers, 1983. [7] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Memoirs of the AMS, 152 (2001), x+108 pp. doi: 10.1090/memo/0722. [8] M. Desbrun, E. Gawlik, F. Gay-Balmaz and V. Zeitlin, Variational discretization for rotating stratified fluids, Disc. Cont. Dyn. Syst. Series A, 34 (2014), 477-509.  doi: 10.3934/dcds.2014.34.477. [9] E. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D, 240 (2011), 1724-1760.  doi: 10.1016/j.physd.2011.07.011. [10] F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Adv. Appl. Math., 42 (2009), 176-275.  doi: 10.1016/j.aam.2008.06.002. [11] F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems, Physica D: Nonlinear Phenomena, 239 (2010), 1929-1947.  doi: 10.1016/j.physd.2010.07.002. [12] F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part Ⅰ: Discrete systems, J. Geom. Phys., 111 (2017), 169-193.  doi: 10.1016/j.geomphys.2016.08.018. [13] F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part Ⅱ: Continuum systems, J. Geom. Phys., 111 (2017), 194-212.  doi: 10.1016/j.geomphys.2016.08.019. [14] F. Gay-Balmaz and H. Yoshimura, Variational discretization for the nonequilibrium thermodynamics of simple systems, Nonlinearity, 31 (2018), 1673-1705.  doi: 10.1088/1361-6544/aaa10e. [15] F. Gay-Balmaz and H. Yoshimura, A variational formulation of nonequilibrium thermodynamics for discrete open systems with mass and heat transfer, Entropy, 20 (2018), Paper No. 163, 26 pp. doi: 10.3390/e20030163. [16] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31, Springer, 2006. [17] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721. [18] D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry, From Finite to Infinite Dimensions, Oxford Texts in Applied and Engineering Mathematics, 12, Oxford University Press, Oxford, 2009. [19] C. Kane, J. E. Marsden and M. Ortiz, Symplectic-energy-momentum preserving variational integrators, J. Math. Phys., 40 (1999), 3353-3371.  doi: 10.1063/1.532892. [20] S. Kim and S. Karrila, Microhydrodynamics: Principles and Selected Applications, Dover, 1991. [21] H. Lamb, Hydrodynamics, 6th revised edition, Cambridge University Press, Cambridge, 1993. [22] M. de León and D. Martín De Diego, Variational integrators and time-dependent Lagrangian systems, Rep. Math. Phys., 49 (2002), 183-192.  doi: 10.1016/S0034-4877(02)80017-9. [23] R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlin. Sci., 16 (2006), 283-328.  doi: 10.1007/s00332-005-0698-1. [24] J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.  doi: 10.1088/0951-7715/12/6/314. [25] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, A Basic Exposition of Classical Mechanical Systems, Second edition. Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. [26] J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, ZAMP, 44 (1993), 17-43.  doi: 10.1007/BF00914351. [27] J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Comm., 1 (1993), 139-164. [28] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [29] D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden and M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D: Nonlinear Phenomena, 240 (2011), 443-458.  doi: 10.1016/j.physd.2010.10.012. [30] A.-T. Petit and P.-L. Dulon, Recherches sur quelques points importants de la théorie de la chaleur, Annales de Chimie et de Physique, 10 (1819), 395-413. [31] R. W. Sharpe, Differential geometry, Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997. [32] E. C. G. Stueckelberg and P. B. Scheurer, Thermocinétique Phénoménologique Galiléenne, Birkhäuser, 1974. [33] V. Zeitlin, Finite-mode analogues of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure, Physica D, 49 (1991), 353-362.  doi: 10.1016/0167-2789(91)90152-Y.
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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