Generalized variational calculus for continuous and discrete mechanical systems

TL;DR

This paper introduces a generalized variational calculus framework that unifies continuous and discrete mechanical systems, including constrained, reduced, and optimal control systems, using tangent and complete lifts adaptable to Lie algebroids.

Contribution

It develops a comprehensive formalism extending variational calculus to various mechanical systems and discrete mechanics, incorporating Lie algebroids and nonholonomic constraints.

Findings

Unified framework for continuous and discrete systems

Extension to Lie algebroids and nonholonomic systems

Applicability to optimal control and constrained systems

Abstract

In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory and so on. This generalized variational calculus is based on two main notions: the tangent lift of curves and the notion of complete lift of a vector field. Both concepts are also adapted for the case of skew-symmetric algebroids, therefore, our formalism easily extends to the case of Lie algebroids and nonholonomic systems. Hence, this framework automatically includes reduced mechanical systems subjected or not to constraints. Finally, we show that our formalism can be used to tackle the case of discrete mechanics, including reduced systems, systems subjected to constraints and discrete optimal control theory.