American Institute of Mathematical Sciences

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

Symplectic groupoids and discrete constrained Lagrangian mechanics

Juan Carlos Marrero 1, , David Martín de Diego 2, and  Ari Stern 3,

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.
Keywords: non-integrable constraints, Discrete Lagrangian mechanics, Lagrangian submanifolds, symplectic groupoids, generating functions, Lie algebroids., Lie groupoids.
Mathematics Subject Classification: Primary: 70G45; Secondary: 53D17, 37M1.
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
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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

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