Variational integrators in discrete vakonomic mechanics

  1. García, Pedro L. 3
  2. Fernández, Antonio 1
  3. Rodrigo, César 2
  1. 1 Department of Applied Mathematics, University of Salamanca,
  2. 2 CINAMIL, Academia Militar, Amadora, Portugal
  3. 3 Department of Mathematics, University of Salamanca
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Revista:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

ISSN: 1578-7303,  1579-1505

Año de publicación: 2012

Volumen: 106

Número: 1

Páginas: 137-159

Tipo: Artículo

DOI: 10.1007/S13398-011-0030-X SCOPUS: 2-s2.0-84856860956 DIALNET GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

Resumen

We introduce the discrete counterpart of the vakonomic method in Lagrangian mechanics with non-holonomic constraints. After defining the concepts of “admissible section” and “admissible infinitesimal variation” of a discrete vakonomic system, we aim to determinate those admissible sections that are critical for the Lagrangian of the system with respect to admissible infinitesimal variations. For sections that satisfy a certain regularity condition, we prove that critical sections are extremals of a variational problem without constraints canonically associated to the initial system (Lagrange multiplier rule). We introduce a notion of “constrained variational integrator”, which is characterized by a Cartan equation that ensures its simplecticity. Moreover, under certain regularity conditions we prove that these integrators can be locally constructed from a generating function of the second kind in the sense of symplectic geometry. Finally, the theory is illustrated with two elementary examples: an isoperimetric problem and an optimal control problem

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