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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
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.  Google Scholar

[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.  Google Scholar

[3]

A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI, 1999. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpretation, Mat. Model., 2 (1990), 78-87.  Google Scholar

[21]

W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114.  Google Scholar

[22]

A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705-727. doi: 10.2969/jmsj/04040705.  Google Scholar

[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.  Google Scholar

[24]

P. Xu, On Poisson groupoids, Internat. J. Math., 6 (1995), 101-124. doi: 10.1142/S0129167X95000080.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI, 1999. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpretation, Mat. Model., 2 (1990), 78-87.  Google Scholar

[21]

W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114.  Google Scholar

[22]

A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705-727. doi: 10.2969/jmsj/04040705.  Google Scholar

[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.  Google Scholar

[24]

P. Xu, On Poisson groupoids, Internat. J. Math., 6 (1995), 101-124. doi: 10.1142/S0129167X95000080.  Google Scholar

[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.  Google Scholar

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