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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 |
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 [
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. |



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