Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Symplectic groupoids and discrete constrained Lagrangian mechanics

Pages: 367 - 397, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.367

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Juan Carlos Marrero - Unidad Asociada ULL-CSIC "Geometría Diferencial y Mecánica Geométrica", Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain (email)
David Martín de Diego - Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, UAM, C/Nicolás Cabrera, 15, 28049 Madrid, Spain (email)
Ari Stern - Department of Mathematics, Washington University in St. Louis, Campus Box 1146, One Brookings Drive, St. Louis, Missouri 63130-4899, United States (email)

Abstract: 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 Śniatycki and Tulczyjew [18]---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.

Keywords:  Discrete Lagrangian mechanics, non-integrable constraints, symplectic groupoids, generating functions, Lagrangian submanifolds, Lie groupoids, Lie algebroids.
Mathematics Subject Classification:  Primary: 70G45; Secondary: 53D17, 37M15.

Received: April 2013;      Revised: July 2014;      Available Online: August 2014.