Morse families and Dirac systems

TL;DR

This paper introduces a geometric framework using Dirac structures and Morse families to unify various mechanics scenarios, providing a generalized approach with an integrability algorithm for implicit differential equations.

Contribution

It generalizes previous Dirac structure results and develops an integrability algorithm for generalized Dirac dynamical systems in mechanics.

Findings

Unified geometric formalism for mechanics scenarios

Extension of Dirac structures to Morse families

An integrability algorithm for implicit differential equations

Abstract

Dirac structures and Morse families are used to obtain a geometric formalism that unifies most of the scenarios in mechanics (constrained calculus, nonholonomic systems, optimal control theory, higher-order mechanics, etc.), as the examples in the paper show. This approach generalizes the previous results on Dirac structures associated with Lagrangian submanifolds. An integrability algorithm in the sense of Mendela, Marmo and Tulczyjew is described for the generalized Dirac dynamical systems under study to determine the set where the implicit differential equations have solutions.