Leibniz-Dirac structures and nonconservative systems with constraints

TL;DR

This paper introduces Leibniz-Dirac structures, a generalization of Dirac structures, enabling the modeling of dissipative systems with constraints, expanding the framework for nonconservative physical systems.

Contribution

The paper defines Leibniz-Dirac structures and characterizes them via bundle maps, extending Dirac structures to include dissipative and gradient systems with constraints.

Findings

Leibniz-Dirac structures generalize Dirac and Riemannian structures.

They facilitate modeling of implicit dissipative Hamiltonian systems.

Physical systems with constraints can be formulated using Leibniz-Dirac structures.

Abstract

Although conservative Hamiltonian systems with constraints can be formulated in terms of Dirac structures, a more general framework is necessary to cover also dissipative systems such as gradient and metriplectic systems with constraints. We define Leibniz-Dirac structures which lead to a natural generalization of Dirac and Riemannian structures, for instance. From modeling point of view, Leibniz-Dirac structures make it easy to formulate implicit dissipative Hamiltonian systems. We give their exact characterization in terms of bundle maps from the tangent bundle to the cotangent bundle and vice verse. Physical systems which can be formulated in terms of Leibniz-Dirac structures are discussed.