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29 March 2024
 
  » arxiv » math.DG/0506299

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Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids
J.C. Marrero ; D. Mart’{i}n de Diego ; E. Mart’{i}nez ;
Date 15 Jun 2005
Subject Differential Geometry; Mathematical Physics MSC-class: 17B66; 22A22; 70G45; 70Hxx | math.DG math-ph math.MP
AbstractThe purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. The equations we get include as particular cases the classical discrete Euler-Lagrange equations, the discrete Euler-Poincaré and discrete Lagrange-Poincaré equations. Our results can be important for the construction of geometric integrators for continuous Lagrangian systems.
Source arXiv, math.DG/0506299
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