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Discrete dynamics in implicit form
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. | 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 |
3. | Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, UAM, C/Nicolás Cabrera, 15, 28049 Madrid, Spain |
References:
[1] |
C. D. Ahlbrandt and A. P. Peterson, "Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, And Riccati Equations," Kluwer Texts in the Mathematical Sciences, Kluwer, Netherlands, 1996. |
[2] |
J. F. Cariñena, Theory of singular lagrangians, Fortschr. Phys., 38 (1990), 641-679.
doi: 10.1002/prop.2190380902. |
[3] |
H. Cendra and M. Etchechoury, Desingularization of implicit analytic differential equations, J. Phys. A, 39 (2006), 10975-11001.
doi: 10.1088/0305-4470/39/35/003. |
[4] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Soc., 152 (2001), x+108 pp. |
[5] |
A. Coste, P. Dazord and A. Weinstein, Grupoï des symplectiques, (French) [Symplectic groupoids], Pub. Dép. Math. Lyon, 2/A (1987), 1-62. |
[6] |
M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620.
doi: 10.4007/annals.2003.157.575. |
[7] |
Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem, SIGMA, 3 (2007), 15 pp. |
[8] |
Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonli-nearity, 18 (2005), 2211-2241. |
[9] |
E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," Springer Series in Computational Mathematics vol. 31, Springer-Verlag, Berlig, 2002. |
[10] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic lagrangian systems on Lie groupoids, J. Nonlinear Sci, 18 (2008), 221-276.
doi: 10.1007/s00332-007-9012-8. |
[11] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular lagrangian systems and variational constrained mechanics on Lie algebroids, Dynamical Systems, 23 (2008), 351-397. |
[12] |
M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308.
doi: 10.1088/0305-4470/38/24/R01. |
[13] |
K. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series vol. 213, Cambridge University Press, Cambridge, 2005. |
[14] |
G. Marmo, G. Mendella and W. M. Tulczyjew, Symmetries and constants of the motion for dynamics in implicit form, Ann. Inst. Henri Poincaré, 57 (1992), 147-166. |
[15] |
G. Marmo, G. Mendella and W. M. Tulczyjew, Integrability of implicit differential equations, J. Phys. A: Math. Gen. 30 (1995), 149-163. |
[16] |
G. Marmo, G. Mendella and W. M. Tulczyjew, Constrained Hamiltonian systems as implicit differential equations, J. Phys. A: Math. Gen., 30 (1997), 277-293.
doi: 10.1088/0305-4470/30/1/020. |
[17] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[18] |
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. |
[19] |
J. C. Marrero, D. Martín de Diego and E. Martínez, The exact discrete lagrangian function on Lie groupoids and some applications, work in progress. |
[20] |
J. C. Marrero, D. Martín de Diego and A. Stern, Lagrangian submanifolds and discrete constrained mechanics on Lie groupoids, preprint, arXiv:1103.6250. |
[21] |
J. Neimark and N. Fufaev, "Dynamics On Nonholonomic Systems," Translation of Mathematics Monographs, vol. 33, AMS, Providence, RI, 1972. |
[22] |
L. Petzold and P. Lötstedt, Numerical solution of nonlinear differential equations with algebraic constraints. II. Practical implications, SIAM J. Sci. Statist. Comput., 7 (1986), 720-733. |
[23] |
P. J. Rabier and W. C. Rheinboldt, Finite difference methods for time dependent, linear differential algebraic equations, Appl. Math. Lett., 7 (1994), 29-34. |
[24] |
J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman & Hall, London 1994 |
[25] |
A. Stern, Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids, J. Symplectic Geom., 8 (2010), 225-238. |
[26] |
W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris, 283 (1976), 15-18. |
[27] |
W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris, 283 (1976), 675-678. |
[28] |
A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705-727.
doi: 10.2969/jmsj/04040705. |
[29] |
A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231. |
show all references
References:
[1] |
C. D. Ahlbrandt and A. P. Peterson, "Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, And Riccati Equations," Kluwer Texts in the Mathematical Sciences, Kluwer, Netherlands, 1996. |
[2] |
J. F. Cariñena, Theory of singular lagrangians, Fortschr. Phys., 38 (1990), 641-679.
doi: 10.1002/prop.2190380902. |
[3] |
H. Cendra and M. Etchechoury, Desingularization of implicit analytic differential equations, J. Phys. A, 39 (2006), 10975-11001.
doi: 10.1088/0305-4470/39/35/003. |
[4] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Soc., 152 (2001), x+108 pp. |
[5] |
A. Coste, P. Dazord and A. Weinstein, Grupoï des symplectiques, (French) [Symplectic groupoids], Pub. Dép. Math. Lyon, 2/A (1987), 1-62. |
[6] |
M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620.
doi: 10.4007/annals.2003.157.575. |
[7] |
Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem, SIGMA, 3 (2007), 15 pp. |
[8] |
Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonli-nearity, 18 (2005), 2211-2241. |
[9] |
E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," Springer Series in Computational Mathematics vol. 31, Springer-Verlag, Berlig, 2002. |
[10] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic lagrangian systems on Lie groupoids, J. Nonlinear Sci, 18 (2008), 221-276.
doi: 10.1007/s00332-007-9012-8. |
[11] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular lagrangian systems and variational constrained mechanics on Lie algebroids, Dynamical Systems, 23 (2008), 351-397. |
[12] |
M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308.
doi: 10.1088/0305-4470/38/24/R01. |
[13] |
K. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series vol. 213, Cambridge University Press, Cambridge, 2005. |
[14] |
G. Marmo, G. Mendella and W. M. Tulczyjew, Symmetries and constants of the motion for dynamics in implicit form, Ann. Inst. Henri Poincaré, 57 (1992), 147-166. |
[15] |
G. Marmo, G. Mendella and W. M. Tulczyjew, Integrability of implicit differential equations, J. Phys. A: Math. Gen. 30 (1995), 149-163. |
[16] |
G. Marmo, G. Mendella and W. M. Tulczyjew, Constrained Hamiltonian systems as implicit differential equations, J. Phys. A: Math. Gen., 30 (1997), 277-293.
doi: 10.1088/0305-4470/30/1/020. |
[17] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[18] |
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. |
[19] |
J. C. Marrero, D. Martín de Diego and E. Martínez, The exact discrete lagrangian function on Lie groupoids and some applications, work in progress. |
[20] |
J. C. Marrero, D. Martín de Diego and A. Stern, Lagrangian submanifolds and discrete constrained mechanics on Lie groupoids, preprint, arXiv:1103.6250. |
[21] |
J. Neimark and N. Fufaev, "Dynamics On Nonholonomic Systems," Translation of Mathematics Monographs, vol. 33, AMS, Providence, RI, 1972. |
[22] |
L. Petzold and P. Lötstedt, Numerical solution of nonlinear differential equations with algebraic constraints. II. Practical implications, SIAM J. Sci. Statist. Comput., 7 (1986), 720-733. |
[23] |
P. J. Rabier and W. C. Rheinboldt, Finite difference methods for time dependent, linear differential algebraic equations, Appl. Math. Lett., 7 (1994), 29-34. |
[24] |
J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman & Hall, London 1994 |
[25] |
A. Stern, Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids, J. Symplectic Geom., 8 (2010), 225-238. |
[26] |
W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris, 283 (1976), 15-18. |
[27] |
W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris, 283 (1976), 675-678. |
[28] |
A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705-727.
doi: 10.2969/jmsj/04040705. |
[29] |
A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231. |
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