# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367
##### References:
 [1] R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0. [2] N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups part I: Introduction and structure-preserving properties, Found. Comput. Math., 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4. [3] A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI, 1999. [4] A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques, 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. [5] Z. Ge, Generating functions, Hamilton-Jacobi equations and symplectic groupoids on Poisson manifolds, Indiana Univ. Math. J., 39 (1990), 859-876. doi: 10.1512/iumj.1990.39.39042. [6] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006,. [7] D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dyn. Syst., 23 (2008), 351-397. doi: 10.1080/14689360802294220. [8] 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. [9] W.-S. Koon and J. E. Marsden, Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction, SIAM J. Control Optim., 35 (1997), 901-929. doi: 10.1137/S0363012995290367. [10] P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Translated from the French by Bertram Eugene Schwarzbach, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. [11] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883. [12] C.-M. Marle, From momentum maps and dual pairs to symplectic and Poisson groupoids, 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. [13] J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348; Corrigendum, Nonlinearity, 19 (2006), 3003-3004. doi: 10.1088/0951-7715/19/6/006. [14] J. C. Marrero, D. Martín de Diego and E. Martínez, The exact discrete Lagrangian function on Lie groupoids and some applications, in preparation. [15] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X. [16] R. I. McLachlan and C. Scovel, A survey of open problems in symplectic integration, in Integration Algorithms and Classical Mechanics (Toronto, ON, 1993), Fields Inst. Commun., 10, Amer. Math. Soc., Providence, RI, 1996, 151-180. [17] J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. [18] J. Śniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J., 22 (1972), 267-275. doi: 10.1512/iumj.1973.22.22021. [19] A. Stern, Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids, J. Symplectic Geom., 8 (2010), 225-238. doi: 10.4310/JSG.2010.v8.n2.a5. [20] Y. B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpretation, Mat. Model., 2 (1990), 78-87. [21] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. [22] A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705-727. doi: 10.2969/jmsj/04040705. [23] A. Weinstein, Lagrangian mechanics and groupoids, in Mechanics Day (Waterloo, ON, 1992), Fields Inst. Commun., 7, American Mathematical Society, Providence, RI, 1996, 207-231. [24] P. Xu, On Poisson groupoids, Internat. J. Math., 6 (1995), 101-124. doi: 10.1142/S0129167X95000080. [25] H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics. II. Variational structures, J. Geom. Phys., 57 (2006), 209-250. doi: 10.1016/j.geomphys.2006.02.012.

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##### References:
 [1] R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0. [2] N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups part I: Introduction and structure-preserving properties, Found. Comput. Math., 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4. [3] A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI, 1999. [4] A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques, 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. [5] Z. Ge, Generating functions, Hamilton-Jacobi equations and symplectic groupoids on Poisson manifolds, Indiana Univ. Math. J., 39 (1990), 859-876. doi: 10.1512/iumj.1990.39.39042. [6] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006,. [7] D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dyn. Syst., 23 (2008), 351-397. doi: 10.1080/14689360802294220. [8] 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. [9] W.-S. Koon and J. E. Marsden, Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction, SIAM J. Control Optim., 35 (1997), 901-929. doi: 10.1137/S0363012995290367. [10] P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Translated from the French by Bertram Eugene Schwarzbach, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. [11] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883. [12] C.-M. Marle, From momentum maps and dual pairs to symplectic and Poisson groupoids, 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. [13] J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348; Corrigendum, Nonlinearity, 19 (2006), 3003-3004. doi: 10.1088/0951-7715/19/6/006. [14] J. C. Marrero, D. Martín de Diego and E. Martínez, The exact discrete Lagrangian function on Lie groupoids and some applications, in preparation. [15] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X. [16] R. I. McLachlan and C. Scovel, A survey of open problems in symplectic integration, in Integration Algorithms and Classical Mechanics (Toronto, ON, 1993), Fields Inst. Commun., 10, Amer. Math. Soc., Providence, RI, 1996, 151-180. [17] J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. [18] J. Śniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J., 22 (1972), 267-275. doi: 10.1512/iumj.1973.22.22021. [19] A. Stern, Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids, J. Symplectic Geom., 8 (2010), 225-238. doi: 10.4310/JSG.2010.v8.n2.a5. [20] Y. B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpretation, Mat. Model., 2 (1990), 78-87. [21] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. [22] A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705-727. doi: 10.2969/jmsj/04040705. [23] A. Weinstein, Lagrangian mechanics and groupoids, in Mechanics Day (Waterloo, ON, 1992), Fields Inst. Commun., 7, American Mathematical Society, Providence, RI, 1996, 207-231. [24] P. Xu, On Poisson groupoids, Internat. J. Math., 6 (1995), 101-124. doi: 10.1142/S0129167X95000080. [25] H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics. II. Variational structures, J. Geom. Phys., 57 (2006), 209-250. doi: 10.1016/j.geomphys.2006.02.012.
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