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March  2013, 33(3): 1117-1135. doi: 10.3934/dcds.2013.33.1117

Discrete dynamics in implicit form

David Iglesias-Ponte 1, , Juan Carlos Marrero 2, , David Martín de Diego 3, and  Edith Padrón 1,

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

Received  April 2011 Revised  November 2011 Published  October 2012

A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid $G$ may be described in terms of Lagrangian implicit difference equations of the corresponding cotangent groupoid $T^*G$. Other situations include finite difference methods for time-dependent linear differential-algebraic equations and discrete nonholonomic Lagrangian systems, as parti-cular examples.
Keywords: Lie groupoids., Discrete mechanics, implicit equations.
Mathematics Subject Classification: Primary: 34K32, 22A22; Secondary: 17B66, 37M15, 57D1.
Citation: David Iglesias-Ponte, Juan Carlos Marrero, David Martín de Diego, Edith Padrón. Discrete dynamics in implicit form. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1117-1135. doi: 10.3934/dcds.2013.33.1117
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.  Google Scholar

[2]

J. F. Cariñena, Theory of singular lagrangians, Fortschr. Phys., 38 (1990), 641-679. doi: 10.1002/prop.2190380902.  Google Scholar

[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.  Google Scholar

[4]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Soc., 152 (2001), x+108 pp.  Google Scholar

[5]

A. Coste, P. Dazord and A. Weinstein, Grupoï des symplectiques, (French) [Symplectic groupoids], Pub. Dép. Math. Lyon, 2/A (1987), 1-62.  Google Scholar

[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.  Google Scholar

[7]

Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem, SIGMA, 3 (2007), 15 pp.  Google Scholar

[8]

Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonli-nearity, 18 (2005), 2211-2241.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

K. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series vol. 213, Cambridge University Press, Cambridge, 2005.  Google Scholar

[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.  Google Scholar

[15]

G. Marmo, G. Mendella and W. M. Tulczyjew, Integrability of implicit differential equations, J. Phys. A: Math. Gen. 30 (1995), 149-163.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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., ().   Google Scholar

[20]

J. C. Marrero, D. Martín de Diego and A. Stern, Lagrangian submanifolds and discrete constrained mechanics on Lie groupoids,, preprint, ().   Google Scholar

[21]

J. Neimark and N. Fufaev, "Dynamics On Nonholonomic Systems," Translation of Mathematics Monographs, vol. 33, AMS, Providence, RI, 1972. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman & Hall, London 1994  Google Scholar

[25]

A. Stern, Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids, J. Symplectic Geom., 8 (2010), 225-238.  Google Scholar

[26]

W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris, 283 (1976), 15-18.  Google Scholar

[27]

W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris, 283 (1976), 675-678.  Google Scholar

[28]

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

[29]

A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.  Google Scholar

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.  Google Scholar

[2]

J. F. Cariñena, Theory of singular lagrangians, Fortschr. Phys., 38 (1990), 641-679. doi: 10.1002/prop.2190380902.  Google Scholar

[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.  Google Scholar

[4]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Soc., 152 (2001), x+108 pp.  Google Scholar

[5]

A. Coste, P. Dazord and A. Weinstein, Grupoï des symplectiques, (French) [Symplectic groupoids], Pub. Dép. Math. Lyon, 2/A (1987), 1-62.  Google Scholar

[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.  Google Scholar

[7]

Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem, SIGMA, 3 (2007), 15 pp.  Google Scholar

[8]

Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonli-nearity, 18 (2005), 2211-2241.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

K. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series vol. 213, Cambridge University Press, Cambridge, 2005.  Google Scholar

[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.  Google Scholar

[15]

G. Marmo, G. Mendella and W. M. Tulczyjew, Integrability of implicit differential equations, J. Phys. A: Math. Gen. 30 (1995), 149-163.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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., ().   Google Scholar

[20]

J. C. Marrero, D. Martín de Diego and A. Stern, Lagrangian submanifolds and discrete constrained mechanics on Lie groupoids,, preprint, ().   Google Scholar

[21]

J. Neimark and N. Fufaev, "Dynamics On Nonholonomic Systems," Translation of Mathematics Monographs, vol. 33, AMS, Providence, RI, 1972. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman & Hall, London 1994  Google Scholar

[25]

A. Stern, Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids, J. Symplectic Geom., 8 (2010), 225-238.  Google Scholar

[26]

W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris, 283 (1976), 15-18.  Google Scholar

[27]

W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris, 283 (1976), 675-678.  Google Scholar

[28]

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

[29]

A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.  Google Scholar

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