American Institute of Mathematical Sciences

Advanced
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 - A, 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, (1996).   Google Scholar

[2]

J. F. Cariñena, Theory of singular lagrangians,, Fortschr. Phys., 38 (1990), 641.  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.  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).   Google Scholar

[5]

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

[6]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Ann. of Math., 157 (2003), 575.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[7]

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

[8]

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

[15]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[2]

J. F. Cariñena, Theory of singular lagrangians,, Fortschr. Phys., 38 (1990), 641.  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.  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).   Google Scholar

[5]

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

[6]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Ann. of Math., 157 (2003), 575.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[7]

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

[8]

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

[15]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[1]

Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367
[2]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017
[3]

Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105
[4]

Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014
[5]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064
[6]

Víctor Manuel Jiménez Morales, Manuel De León, Marcelo Epstein. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. Journal of Geometric Mechanics, 2019, 11 (3) : 301-324. doi: 10.3934/jgm.2019017
[7]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421
[8]

P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1
[9]

Farid Tari. Two-parameter families of implicit differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 139-162. doi: 10.3934/dcds.2005.13.139
[10]

Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87
[11]

Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009
[12]

Firas Hindeleh, Gerard Thompson. Killing's equations for invariant metrics on Lie groups. Journal of Geometric Mechanics, 2011, 3 (3) : 323-335. doi: 10.3934/jgm.2011.3.323
[13]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[14]

Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829
[15]

Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549
[16]

Andrew N. W. Hone, Matteo Petrera. Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras. Journal of Geometric Mechanics, 2009, 1 (1) : 55-85. doi: 10.3934/jgm.2009.1.55
[17]

Fernando Casas, Cristina Chiralt. A Lie--Deprit perturbation algorithm for linear differential equations with periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 959-975. doi: 10.3934/dcds.2014.34.959
[18]

Luca Capogna. Optimal regularity for quasilinear equations in stratified nilpotent Lie groups and applications. Electronic Research Announcements, 1996, 2: 60-68.

[19]

Isaac A. García, Jaume Giné, Jaume Llibre. Liénard and Riccati differential equations related via Lie Algebras. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 485-494. doi: 10.3934/dcdsb.2008.10.485
[20]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences