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Symplectic groupoids and discrete constrained Lagrangian mechanics. (English) Zbl 1305.70032
Summary: In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions. First, we show that Lagrangian submanifolds of symplectic groupoids give rise to discrete dynamical systems, and we study the properties of these systems, including their regularity and reversibility, from the perspective of symplectic and Poisson geometry. Next, we use this framework – along with a generalized notion of generating function due to J. Śniatycki and W. M. Tulczyjew [Indiana Univ. Math. J. 22, 267–275 (1972; Zbl 0237.58002)] – to develop a theory of discrete constrained Lagrangian mechanics. This allows for systems with arbitrary constraints, including those which are non-integrable (in an appropriate discrete, variational sense). In addition to characterizing the dynamics of these constrained systems, we also develop a theory of reduction and Noether symmetries, and study the relationship between the dynamics and variational principles. Finally, we apply this theory to discretize several concrete examples of constrained systems in mechanics and optimal control.

MSC:
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
53D17 Poisson manifolds; Poisson groupoids and algebroids
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
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