January  2015, 35(1): 367-397. doi: 10.3934/dcds.2015.35.367

Symplectic groupoids and discrete constrained Lagrangian mechanics

1. 

Unidad Asociada ULL-CSIC "Geometría Diferencial y Mecánica Geométrica", Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain

2. 

Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, UAM, C/Nicolás Cabrera, 15, 28049 Madrid

3. 

Department of Mathematics, Washington University in St. Louis, Campus Box 1146, One Brookings Drive, St. Louis, Missouri 63130-4899, United States

Received  April 2013 Revised  July 2014 Published  August 2014

In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions. First, we show that Lagrangian submanifolds of symplectic groupoids give rise to discrete dynamical systems, and we study the properties of these systems, including their regularity and reversibility, from the perspective of symplectic and Poisson geometry. Next, we use this framework---along with a generalized notion of generating function due to Śniatycki and Tulczyjew [18]---to develop a theory of discrete constrained Lagrangian mechanics. This allows for systems with arbitrary constraints, including those which are non-integrable (in an appropriate discrete, variational sense). In addition to characterizing the dynamics of these constrained systems, we also develop a theory of reduction and Noether symmetries, and study the relationship between the dynamics and variational principles. Finally, we apply this theory to discretize several concrete examples of constrained systems in mechanics and optimal control.
Citation: Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367
References:
[1]

2nd edition, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar

[2]

Found. Comput. Math., 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4.  Google Scholar

[3]

Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI, 1999. Google Scholar

[4]

in Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, 87, Univ. Claude-Bernard, Lyon, 1987, i-ii, 1-62.  Google Scholar

[5]

Indiana Univ. Math. J., 39 (1990), 859-876. doi: 10.1512/iumj.1990.39.39042.  Google Scholar

[6]

2nd edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006,. Google Scholar

[7]

Dyn. Syst., 23 (2008), 351-397. doi: 10.1080/14689360802294220.  Google Scholar

[8]

J. Math. Phys., 40 (1999), 3353-3371. doi: 10.1063/1.532892.  Google Scholar

[9]

SIAM J. Control Optim., 35 (1997), 901-929. doi: 10.1137/S0363012995290367.  Google Scholar

[10]

Translated from the French by Bertram Eugene Schwarzbach, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. Google Scholar

[11]

London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[12]

in The Breadth of Symplectic and Poisson Geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 493-523. doi: 10.1007/0-8176-4419-9\_17.  Google Scholar

[13]

Nonlinearity, 19 (2006), 1313-1348; Corrigendum, Nonlinearity, 19 (2006), 3003-3004. doi: 10.1088/0951-7715/19/6/006.  Google Scholar

[14]

J. C. Marrero, D. Martín de Diego and E. Martínez, The exact discrete Lagrangian function on Lie groupoids and some applications,, in preparation., ().   Google Scholar

[15]

Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X.  Google Scholar

[16]

in Integration Algorithms and Classical Mechanics (Toronto, ON, 1993), Fields Inst. Commun., 10, Amer. Math. Soc., Providence, RI, 1996, 151-180.  Google Scholar

[17]

Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.  Google Scholar

[18]

Indiana Univ. Math. J., 22 (1972), 267-275. doi: 10.1512/iumj.1973.22.22021.  Google Scholar

[19]

J. Symplectic Geom., 8 (2010), 225-238. doi: 10.4310/JSG.2010.v8.n2.a5.  Google Scholar

[20]

Mat. Model., 2 (1990), 78-87.  Google Scholar

[21]

Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114.  Google Scholar

[22]

J. Math. Soc. Japan, 40 (1988), 705-727. doi: 10.2969/jmsj/04040705.  Google Scholar

[23]

in Mechanics Day (Waterloo, ON, 1992), Fields Inst. Commun., 7, American Mathematical Society, Providence, RI, 1996, 207-231.  Google Scholar

[24]

Internat. J. Math., 6 (1995), 101-124. doi: 10.1142/S0129167X95000080.  Google Scholar

[25]

J. Geom. Phys., 57 (2006), 209-250. doi: 10.1016/j.geomphys.2006.02.012.  Google Scholar

show all references

References:
[1]

2nd edition, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar

[2]

Found. Comput. Math., 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4.  Google Scholar

[3]

Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI, 1999. Google Scholar

[4]

in Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, 87, Univ. Claude-Bernard, Lyon, 1987, i-ii, 1-62.  Google Scholar

[5]

Indiana Univ. Math. J., 39 (1990), 859-876. doi: 10.1512/iumj.1990.39.39042.  Google Scholar

[6]

2nd edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006,. Google Scholar

[7]

Dyn. Syst., 23 (2008), 351-397. doi: 10.1080/14689360802294220.  Google Scholar

[8]

J. Math. Phys., 40 (1999), 3353-3371. doi: 10.1063/1.532892.  Google Scholar

[9]

SIAM J. Control Optim., 35 (1997), 901-929. doi: 10.1137/S0363012995290367.  Google Scholar

[10]

Translated from the French by Bertram Eugene Schwarzbach, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. Google Scholar

[11]

London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[12]

in The Breadth of Symplectic and Poisson Geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 493-523. doi: 10.1007/0-8176-4419-9\_17.  Google Scholar

[13]

Nonlinearity, 19 (2006), 1313-1348; Corrigendum, Nonlinearity, 19 (2006), 3003-3004. doi: 10.1088/0951-7715/19/6/006.  Google Scholar

[14]

J. C. Marrero, D. Martín de Diego and E. Martínez, The exact discrete Lagrangian function on Lie groupoids and some applications,, in preparation., ().   Google Scholar

[15]

Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X.  Google Scholar

[16]

in Integration Algorithms and Classical Mechanics (Toronto, ON, 1993), Fields Inst. Commun., 10, Amer. Math. Soc., Providence, RI, 1996, 151-180.  Google Scholar

[17]

Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.  Google Scholar

[18]

Indiana Univ. Math. J., 22 (1972), 267-275. doi: 10.1512/iumj.1973.22.22021.  Google Scholar

[19]

J. Symplectic Geom., 8 (2010), 225-238. doi: 10.4310/JSG.2010.v8.n2.a5.  Google Scholar

[20]

Mat. Model., 2 (1990), 78-87.  Google Scholar

[21]

Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114.  Google Scholar

[22]

J. Math. Soc. Japan, 40 (1988), 705-727. doi: 10.2969/jmsj/04040705.  Google Scholar

[23]

in Mechanics Day (Waterloo, ON, 1992), Fields Inst. Commun., 7, American Mathematical Society, Providence, RI, 1996, 207-231.  Google Scholar

[24]

Internat. J. Math., 6 (1995), 101-124. doi: 10.1142/S0129167X95000080.  Google Scholar

[25]

J. Geom. Phys., 57 (2006), 209-250. doi: 10.1016/j.geomphys.2006.02.012.  Google Scholar

[1]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421

[2]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

[3]

Víctor Manuel Jiménez Morales, Manuel De León, Marcelo Epstein. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. Journal of Geometric Mechanics, 2019, 11 (3) : 301-324. doi: 10.3934/jgm.2019017

[4]

Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009

[5]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064

[6]

Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811

[7]

Gennadi Sardanashvily. Lagrangian dynamics of submanifolds. Relativistic mechanics. Journal of Geometric Mechanics, 2012, 4 (1) : 99-110. doi: 10.3934/jgm.2012.4.99

[8]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017

[9]

Dennise García-Beltrán, José A. Vallejo, Yurii Vorobiev. Lie algebroids generated by cohomology operators. Journal of Geometric Mechanics, 2015, 7 (3) : 295-315. doi: 10.3934/jgm.2015.7.295

[10]

Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211

[11]

Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105

[12]

Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014

[13]

K. C. H. Mackenzie. Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids. Electronic Research Announcements, 1998, 4: 74-87.

[14]

Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004

[15]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81

[16]

Cédric M. Campos, Elisa Guzmán, Juan Carlos Marrero. Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (1) : 1-26. doi: 10.3934/jgm.2012.4.1

[17]

Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 67-98. doi: 10.3934/jgm.2014.6.67

[18]

Marco Zambon, Chenchang Zhu. Distributions and quotients on degree $1$ NQ-manifolds and Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 469-485. doi: 10.3934/jgm.2012.4.469

[19]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

[20]

Juan Carlos Marrero. Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes. Journal of Geometric Mechanics, 2010, 2 (3) : 243-263. doi: 10.3934/jgm.2010.2.243

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (5)

[Back to Top]