The Journal of Geometric Mechanics (JGM)

Variational integrators for discrete Lagrange problems

Pages: 343 - 374, Volume 2, Issue 4, December 2010      doi:10.3934/jgm.2010.2.343

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Pedro L. García - Department of Mathematics, University of Salamanca, Salamanca 37008, Spain (email)
Antonio Fernández - Department of Applied Mathematics, University of Salamanca, Salamanca 37008, Spain (email)
César Rodrigo - CINAMIL, Academia Militar, Amadora 2720-113, Portugal (email)

Abstract: 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:  Variational integrators, discrete Lagrange problems, discrete Cartan form, generating functions.
Mathematics Subject Classification:  37J60, 37M15, 49J15, 65P10.

Received: August 2010;      Revised: December 2010;      Available Online: January 2011.