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

Stern, Ari

J. Symplectic Geom. 8, No. 2, 225-238 (2010).

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

Keywords:

discrete analog of Hamilton-Pontryagin variational principle; discrete Lagrangian mechanics; Lie groupoid; Hamilton’s action principle; symplectic generating function

Citations:

Zbl 1121.53057