Symplectic groupoids and discrete constrained Lagrangian mechanics

TL;DR

This paper extends discrete Lagrangian mechanics using symplectic groupoids, enabling the analysis of constrained systems with symmetries and their discretization in mechanics and control.

Contribution

It introduces a novel framework linking symplectic groupoids with discrete constrained Lagrangian mechanics, including reduction and symmetry analysis.

Findings

Discrete dynamical systems from Lagrangian submanifolds are characterized.

The framework handles non-integrable constraints in discrete systems.

Applications include discretization of constrained mechanical and control systems.

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 Sniatycki and Tulczyjew -- 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.