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Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids

J. C. Marrero·D. Martín de Diego·E. Martínez
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Abstract

The 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.

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J. C. Marrero, D. Martín de Diego, E. Martínez. 2006-11-27. Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. https://doi.org/10.1088/0951-7715%2F19%2F6%2F006

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