Variational integrators for discrete Lagrange problems

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December 2010, 2(4): 343-374. doi: 10.3934/jgm.2010.2.343

Variational integrators for discrete Lagrange problems

Pedro L. García ^1, , Antonio Fernández ^2, and César Rodrigo ^3,

1.	Department of Mathematics, University of Salamanca, Salamanca 37008, Spain
2.	Department of Applied Mathematics, University of Salamanca, Salamanca 37008, Spain
3.	CINAMIL, Academia Militar, Amadora 2720-113, Portugal

Received August 2010 Revised December 2010 Published January 2011

Abstract
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A discrete Lagrange problem is defined as a discrete Lagrangian system endowed with a constraint submanifold in the space of 1-jets of the discrete fibred manifold that configures the system. After defining the concepts of admissible section and infinitesimal admissible variation, the objective of these problems is to find admissible sections that are critical for the Lagrangian of the system with respect to the infinitesimal admissible variations. For admissible sections satisfying a certain regularity condition, we prove that critical sections are the solutions of an extended unconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). Next, we define the concept of Cartan 1-form, establish a Noether theory for symmetries and introduce a notion of "constrained variational integrator" that we characterize through a Cartan equation ensuring its symplecticity. Under a certain regularity condition of the problem of Lagrange, we prove the existence and uniqueness of this kind of integrators in the neighborhood of a critical section, showing then that such integrators can be constructed from a generating function of the second class in the sense of symplectic geometry. Finally, the whole theory is illustrated with three elementary examples.

Keywords: generating functions., discrete Lagrange problems, discrete Cartan form, Variational integrators.

Mathematics Subject Classification: 37J60, 37M15, 49J15, 65P1.

Citation: Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343

References:

[1]	V. I. Arnol'd, V. V. Kozlov and A. I. Neĭshtadt, "Dynamical Systems III," Encyclopaedia of Mathematical Sciences, 3, Springer Verlag, Berlin, 1988.
[2]	R. Benito and D. Martín de Diego, Discrete vakonomic mechanics, J. Math. Phys., 46 (2005), 083521 doi: 10.1063/1.2008214.
[3]	A. M. Bloch, "Nonholonomic Mechanics and Control,'' Interdisciplinary Applied Mathematics, 24. Springer Science Ed., 2003.
[4]	F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18 (1996), 295-325. doi: 10.1016/0393-0440(95)00016-X.
[5]	J.-B. Chen, H.-Y. Guo and K. Wu, Total variation and variational symplectic-energy-momentum integrators, preprint, arXiv:hep-th/0109178.
[6]	J.-B. Chen, H.-Y. Guo and K. Wu, Discrete total variation calculus and Lee's discrete mechanics, Appl. Math. Comput., 177 (2006), 226-234. doi: 10.1016/j.amc.2005.11.002.
[7]	J. Cortés, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'' Lect. Notes in Math. 1793, Springer Verlag, 2002.
[8]	P. L. García, A. García and C. Rodrigo, Cartan forms for first order constrained variational problems, J. Geom. Phys., 56 (2006), 571-610. doi: 10.1016/j.geomphys.2005.04.002.
[9]	P. L. García and C. Rodrigo, Cartan forms and second variation for constrained variational problems, Proceedings of the VII International Conference on Geometry, Integrability and Quantization (Varna, Bulgary) 7 Bulgarian Acad. Sci., Sofia, (2006), 140-153.
[10]	H. Goldstein, "Classical Mechanics,'' Addison-Wesley Series in Physics, 1980.
[11]	X. Gràcia, J. Marín Solano and M. C. Muñoz Lecanda, Some geometric aspects of variational calculus in constrained systems, Rep. Math. Phys., 51 (2003), 127-148 doi: 10.1016/S0034-4877(03)80006-X.
[12]	V. M. Guibout and A. Bloch, Discrete variational principles and Hamilton-Jacobi theory for mechanical systems and optimal control problems, e-print ccsd-00002863, version1-2004
[13]	L. Hsu, Calculus of variations via the Griffiths formalism, J. Diff. Geom., 36 (1992), 551-589.
[14]	T. D. Lee, Can time be a discrete dynamical variable?, Phys. Lett. B, 122 (1983), 217-–220. doi: 10.1016/0370-2693(83)90687-1.
[15]	M. de León, D. Martín de Diego and A. Santamaría Merino, Geometric integrators and nonholonomic mechanics, J. Math. Phys., 45 (2004).
[16]	M. de León, D. Martín de Diego and A. Santamaría Merino, Discrete variational integrators and optimal control theory, Advances in Computational Mathematics, 26 (2006), 251-268
[17]	M. de León, J. C. Marrero and D. Martín de Diego, Vakonomic mechanics versus non-holonomic mechanics: A unified geometrical approach, J. Geom. Phys., 35 (2000), 126-144. doi: 10.1016/S0393-0440(00)00004-8.
[18]	J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDEs, Comm. in Math. Phys., 199 (1998), 351-398.
[19]	J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 317-514. doi: 10.1017/S096249290100006X.
[20]	S. Martínez, J. Cortés and M. de León, Symmetries in vakonomic dynamics: Applications to optimal control, J. Geom. Phys., 38 (2001), 343-365. doi: 10.1016/S0393-0440(00)00069-3.
[21]	P. Piccione and D. V. Tausk, Lagrangian and Hamiltonian formalism for constrained variational problems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1417-1437.
[22]	J. Vankerschaver, F. Cantrijn, M. de León and D. Martín de Diego, Geometric aspects of nonholonomic field theories, Rep. Math. Phys., 56 (2005), 387-411. doi: 10.1016/S0034-4877(05)80093-X.
[23]	J. Vankerschaver and F. Cantrijn, Discrete Lagrangian field theories on Lie groupoids, J. Geom. Phys., 57 (2007), 665-689. doi: 10.1016/j.geomphys.2006.05.006.
[24]	M. West, "Variational Integrators,'' Ph.D. Thesis, California Institute of Technology, 2004.

show all references

References:

[1]	V. I. Arnol'd, V. V. Kozlov and A. I. Neĭshtadt, "Dynamical Systems III," Encyclopaedia of Mathematical Sciences, 3, Springer Verlag, Berlin, 1988.
[2]	R. Benito and D. Martín de Diego, Discrete vakonomic mechanics, J. Math. Phys., 46 (2005), 083521 doi: 10.1063/1.2008214.
[3]	A. M. Bloch, "Nonholonomic Mechanics and Control,'' Interdisciplinary Applied Mathematics, 24. Springer Science Ed., 2003.
[4]	F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18 (1996), 295-325. doi: 10.1016/0393-0440(95)00016-X.
[5]	J.-B. Chen, H.-Y. Guo and K. Wu, Total variation and variational symplectic-energy-momentum integrators, preprint, arXiv:hep-th/0109178.
[6]	J.-B. Chen, H.-Y. Guo and K. Wu, Discrete total variation calculus and Lee's discrete mechanics, Appl. Math. Comput., 177 (2006), 226-234. doi: 10.1016/j.amc.2005.11.002.
[7]	J. Cortés, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'' Lect. Notes in Math. 1793, Springer Verlag, 2002.
[8]	P. L. García, A. García and C. Rodrigo, Cartan forms for first order constrained variational problems, J. Geom. Phys., 56 (2006), 571-610. doi: 10.1016/j.geomphys.2005.04.002.
[9]	P. L. García and C. Rodrigo, Cartan forms and second variation for constrained variational problems, Proceedings of the VII International Conference on Geometry, Integrability and Quantization (Varna, Bulgary) 7 Bulgarian Acad. Sci., Sofia, (2006), 140-153.
[10]	H. Goldstein, "Classical Mechanics,'' Addison-Wesley Series in Physics, 1980.
[11]	X. Gràcia, J. Marín Solano and M. C. Muñoz Lecanda, Some geometric aspects of variational calculus in constrained systems, Rep. Math. Phys., 51 (2003), 127-148 doi: 10.1016/S0034-4877(03)80006-X.
[12]	V. M. Guibout and A. Bloch, Discrete variational principles and Hamilton-Jacobi theory for mechanical systems and optimal control problems, e-print ccsd-00002863, version1-2004
[13]	L. Hsu, Calculus of variations via the Griffiths formalism, J. Diff. Geom., 36 (1992), 551-589.
[14]	T. D. Lee, Can time be a discrete dynamical variable?, Phys. Lett. B, 122 (1983), 217-–220. doi: 10.1016/0370-2693(83)90687-1.
[15]	M. de León, D. Martín de Diego and A. Santamaría Merino, Geometric integrators and nonholonomic mechanics, J. Math. Phys., 45 (2004).
[16]	M. de León, D. Martín de Diego and A. Santamaría Merino, Discrete variational integrators and optimal control theory, Advances in Computational Mathematics, 26 (2006), 251-268
[17]	M. de León, J. C. Marrero and D. Martín de Diego, Vakonomic mechanics versus non-holonomic mechanics: A unified geometrical approach, J. Geom. Phys., 35 (2000), 126-144. doi: 10.1016/S0393-0440(00)00004-8.
[18]	J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDEs, Comm. in Math. Phys., 199 (1998), 351-398.
[19]	J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 317-514. doi: 10.1017/S096249290100006X.
[20]	S. Martínez, J. Cortés and M. de León, Symmetries in vakonomic dynamics: Applications to optimal control, J. Geom. Phys., 38 (2001), 343-365. doi: 10.1016/S0393-0440(00)00069-3.
[21]	P. Piccione and D. V. Tausk, Lagrangian and Hamiltonian formalism for constrained variational problems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1417-1437.
[22]	J. Vankerschaver, F. Cantrijn, M. de León and D. Martín de Diego, Geometric aspects of nonholonomic field theories, Rep. Math. Phys., 56 (2005), 387-411. doi: 10.1016/S0034-4877(05)80093-X.
[23]	J. Vankerschaver and F. Cantrijn, Discrete Lagrangian field theories on Lie groupoids, J. Geom. Phys., 57 (2007), 665-689. doi: 10.1016/j.geomphys.2006.05.006.
[24]	M. West, "Variational Integrators,'' Ph.D. Thesis, California Institute of Technology, 2004.

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