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Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids. (English) Zbl 1375.70057

Summary: We present a discrete analog of the recently introduced Hamilton-Pontryagin variational principle in Lagrangian mechanics [H. Yoshimura and J. E. Marsden, J. Geom. Phys. 57, No. 1, 209–250 (2006; Zbl 1121.53057)]. This unifies two, previously disparate approaches to discrete Lagrangian mechanics: either using the discrete Lagrangian to define a finite version of Hamilton’s action principle, or treating it as a symplectic generating function. This is demonstrated for a discrete Lagrangian defined on an arbitrary Lie groupoid; the often encountered special case of the pair groupoid (or Cartesian square) is also given as a worked example.

MSC:

70H03 Lagrange’s equations
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
58H05 Pseudogroups and differentiable groupoids

Citations:

Zbl 1121.53057
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