Dirac Algebroids in Lagrangian and Hamiltonian Mechanics

TL;DR

This paper introduces Dirac algebroids as a unified geometric framework for constrained Lagrangian and Hamiltonian systems, encompassing both regular and singular cases, with applications to systems on Lie algebroids.

Contribution

It develops the concept of Dirac algebroids, unifying the geometric treatment of constrained implicit Lagrangian and Hamiltonian systems in an intrinsic way.

Findings

Provides a geometric formulation for all types of Lagrangian and Hamiltonian systems.

Includes a treatment of systems with nonholonomic constraints.

Establishes foundational properties of Dirac and Dirac-Lie algebroids.

Abstract

We present a unified approach to constrained implicit Lagrangian and Hamiltonian systems based on the introduced concept of Dirac algebroid. The latter is a certain almost Dirac structure associated with the Courant algebroid on the dual $E^\ast$ to a vector bundle $E$. If this almost Dirac structure is integrable (Dirac), we speak about a Dirac-Lie algebroid. The bundle $E$ plays the role of the bundle of kinematic configurations (quasi-velocities), while the bundle $E^\ast$ - the role of the phase space. This setting is totally intrinsic and does not distinguish between regular and singular Lagrangians. The constraints are part of the framework, so the general approach does not change when nonholonomic constraints are imposed, and produces the (implicit) Euler-Lagrange and Hamilton equations in an elegant geometric way. The scheme includes all important cases of Lagrangian and Hamiltonian systems, no matter if they are with or without constraints, autonomous or non-autonomous etc., as well as their reductions; in particular, constrained systems on Lie algebroids. we prove also some basic facts about the geometry of Dirac and Dirac-Lie algebroids.