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Contribution to the ergodic theory of robustly transitive maps
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 |
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications,, 2nd edition, (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.
doi: 10.1007/s10208-008-9030-4. |
[3] |
A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras,, Berkeley Mathematics Lecture Notes, (1999). Google Scholar |
[4] |
A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques,, in Publications du Département de Mathématiques. Nouvelle Série. A, (1987), 1.
|
[5] |
Z. Ge, Generating functions, Hamilton-Jacobi equations and symplectic groupoids on Poisson manifolds,, Indiana Univ. Math. J., 39 (1990), 859.
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, (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.
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.
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.
doi: 10.1137/S0363012995290367. |
[10] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics,, Translated from the French by Bertram Eugene Schwarzbach, (1987). Google Scholar |
[11] |
K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical Society Lecture Note Series, (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, (2005), 493.
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.
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., (). Google Scholar |
[15] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357.
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, (1993), 151.
|
[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.
doi: 10.1007/BF02352494. |
[18] |
J. Śniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds,, Indiana Univ. Math. J., 22 (1972), 267.
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.
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.
|
[21] |
W. M. Tulczyjew, The Legendre transformation,, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101.
|
[22] |
A. Weinstein, Coisotropic calculus and Poisson groupoids,, J. Math. Soc. Japan, 40 (1988), 705.
doi: 10.2969/jmsj/04040705. |
[23] |
A. Weinstein, Lagrangian mechanics and groupoids,, in Mechanics Day (Waterloo, (1992), 207.
|
[24] |
P. Xu, On Poisson groupoids,, Internat. J. Math., 6 (1995), 101.
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.
doi: 10.1016/j.geomphys.2006.02.012. |
show all references
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications,, 2nd edition, (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.
doi: 10.1007/s10208-008-9030-4. |
[3] |
A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras,, Berkeley Mathematics Lecture Notes, (1999). Google Scholar |
[4] |
A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques,, in Publications du Département de Mathématiques. Nouvelle Série. A, (1987), 1.
|
[5] |
Z. Ge, Generating functions, Hamilton-Jacobi equations and symplectic groupoids on Poisson manifolds,, Indiana Univ. Math. J., 39 (1990), 859.
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, (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.
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.
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.
doi: 10.1137/S0363012995290367. |
[10] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics,, Translated from the French by Bertram Eugene Schwarzbach, (1987). Google Scholar |
[11] |
K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical Society Lecture Note Series, (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, (2005), 493.
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.
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., (). Google Scholar |
[15] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357.
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, (1993), 151.
|
[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.
doi: 10.1007/BF02352494. |
[18] |
J. Śniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds,, Indiana Univ. Math. J., 22 (1972), 267.
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.
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.
|
[21] |
W. M. Tulczyjew, The Legendre transformation,, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101.
|
[22] |
A. Weinstein, Coisotropic calculus and Poisson groupoids,, J. Math. Soc. Japan, 40 (1988), 705.
doi: 10.2969/jmsj/04040705. |
[23] |
A. Weinstein, Lagrangian mechanics and groupoids,, in Mechanics Day (Waterloo, (1992), 207.
|
[24] |
P. Xu, On Poisson groupoids,, Internat. J. Math., 6 (1995), 101.
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.
doi: 10.1016/j.geomphys.2006.02.012. |
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