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

Variational discretization of thermodynamical simple systems on Lie groups

Benjamin Couéraud 1, and  François Gay-Balmaz 2,,

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.

Keywords: Variational integrators, structure preserving discretization, entropy, nonequilibrium thermodynamics, discrete Lagrangian formulation.
Mathematics Subject Classification: Primary: 37D35, 37M15; Secondary: 49S05, 80M30.
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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

H. Brenner and J. Happel, Low Reynolds Number Hydrodynamics, Mechanics of fluids and transport processes, 1, Martinus Nijhoff publishers, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31, Springer, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

S. Kim and S. Karrila, Microhydrodynamics: Principles and Selected Applications, Dover, 1991. Google Scholar

[21]

H. Lamb, Hydrodynamics, 6th revised edition, Cambridge University Press, Cambridge, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, ZAMP, 44 (1993), 17-43.  doi: 10.1007/BF00914351.  Google Scholar

[27]

J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Comm., 1 (1993), 139-164.   Google Scholar

[28]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[31]

R. W. Sharpe, Differential geometry, Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997.  Google Scholar

[32]

E. C. G. Stueckelberg and P. B. Scheurer, Thermocinétique Phénoménologique Galiléenne, Birkhäuser, 1974.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

H. Brenner and J. Happel, Low Reynolds Number Hydrodynamics, Mechanics of fluids and transport processes, 1, Martinus Nijhoff publishers, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31, Springer, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

S. Kim and S. Karrila, Microhydrodynamics: Principles and Selected Applications, Dover, 1991. Google Scholar

[21]

H. Lamb, Hydrodynamics, 6th revised edition, Cambridge University Press, Cambridge, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, ZAMP, 44 (1993), 17-43.  doi: 10.1007/BF00914351.  Google Scholar

[27]

J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Comm., 1 (1993), 139-164.   Google Scholar

[28]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[31]

R. W. Sharpe, Differential geometry, Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997.  Google Scholar

[32]

E. C. G. Stueckelberg and P. B. Scheurer, Thermocinétique Phénoménologique Galiléenne, Birkhäuser, 1974.  Google Scholar

[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.  Google Scholar

Figure 1.  The trajectory of the center of mass of the heavy top
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Figure 2.  The different energies of the system
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Figure 3.  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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