Variational integrators for discrete Lagrange problems

  1. L. García, Pedro 1
  2. Fernández, Antonio 3
  3. Rodrigo, César 2
  1. 1 Department of Mathematics, University of Salamanca
  2. 2 CINAMIL, Academia Militar, Amadora 2720-113
  3. 3 Department of Applied Mathematics, University of Salamanca
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Revue:
Journal of Geometric Mechanics

ISSN: 1941-4897

Année de publication: 2010

Volumen: 2

Número: 4

Pages: 343-374

Type: Article

DOI: 10.3934/JGM.2010.2.343 GOOGLE SCHOLAR lock_openAccès ouvert editor

D'autres publications dans: Journal of Geometric Mechanics

Résumé

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.

Références bibliographiques

  • V. I. Arnol'd, (1988), Encyclopaedia of Mathematical Sciences, 3, 10.1007/978-3-662-02535-2
  • R. Benito, (2005), J. Math. Phys., 46, 10.1063/1.2008214
  • A. M. Bloch, (2003), Interdisciplinary Applied Mathematics, 24, 10.1007/b97376_6
  • F. Cardin, (1996), J. Geom. Phys., 18, pp. 295, 10.1016/0393-0440(95)00016-X
  • J.-B. Chen, preprint
  • J.-B. Chen, (2006), Appl. Math. Comput., 177, pp. 226, 10.1016/j.amc.2005.11.002
  • J. Cortés, (2002), Lect. Notes in Math. \textbf{1793}, 1793
  • P. L. García, (2006), J. Geom. Phys., 56, pp. 571, 10.1016/j.geomphys.2005.04.002
  • P. L. García, (2006), Proceedings of the VII International Conference on Geometry, pp. 140
  • H. Goldstein, (1980), Addison-Wesley Series in Physics
  • X. Gràcia, (2003), Rep. Math. Phys., 51, pp. 127, 10.1016/S0034-4877(03)80006-X
  • V. M. Guibout, e-print ccsd-00002863, pp. 1
  • L. Hsu, (1992), J. Diff. Geom., 36, pp. 551, 10.4310/jdg/1214453181
  • T. D. Lee, (1983), Phys. Lett. B, 122, 10.1016/0370-2693(83)90687-1
  • M. de León, (2004), J. Math. Phys., 45, 10.1063/1.1644325
  • M. de León, (2006), Advances in Computational Mathematics, 26, pp. 251, 10.1007/s10444-004-4093-5
  • M. de León, (2000), J. Geom. Phys., 35, pp. 126, 10.1016/S0393-0440(00)00004-8
  • J. E. Marsden, (1998), Comm. in Math. Phys., 199, pp. 351, 10.1007/s002200050505
  • J. E. Marsden, (2001), Acta Numerica, 10, pp. 317, 10.1017/S096249290100006X
  • S. Martínez, (2001), J. Geom. Phys., 38, pp. 343, 10.1016/S0393-0440(00)00069-3
  • P. Piccione, (2002), Proc. Roy. Soc. Edinburgh Sect. A, 132, pp. 1417, 10.1017/S0308210500002183
  • J. Vankerschaver, (2005), Rep. Math. Phys., 56, pp. 387, 10.1016/S0034-4877(05)80093-X
  • J. Vankerschaver, (2007), J. Geom. Phys., 57, pp. 665, 10.1016/j.geomphys.2006.05.006
  • M. West, (2004), Ph.D. Thesis