2015/06/29 by Leonardo Colombo, David Martı́n de Diego, Colombo, Leonardo +2
Mathematics · Medicine · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Optimization and Control (math.OC) #Spinal Hematomas and Complications #Symplectic Geometry (math.SG) #math-ph #math.DS #math.MP #math.OC #math.SG
paper · pdf · doi:10.48550/arxiv.1506.08580
41 pages, 1 figure, first version. Comments welcome
arxiv created 2015/06/29 · openalex publication_date 2015/06/29 · arxiv updated 2015/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques for second-order variational problems on Lie groupoids and their applications to the construction of variational integrators for optimal control problems of mechanical systems. Next, we show how Lagrangian submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we study the geometric properties of the implicit flow which defines the dynamics in the Lagrangian submanifold. We also study the theory of reduction by symmetries and the corresponding Noether theorem.