The contact Eden bracket and the evolution of observables

TL;DR

The paper extends nonholonomic mechanics into a contact-geometry setting to address systems with dissipation. It introduces the contact Eden bracket by projecting dynamics to a constraint submanifold via a projection $\gamma$, establishing that the evolution of an observable $f$ in the constrained system corresponds to the evolution of $f\circ\gamma$ in the unconstrained system. A key result is that, for observables satisfying a mechanical condition, the Eden bracket reduces to the canonical Jacobi bracket, enabling an unconstrained-like evolution on a subspace of observables. Additionally, the framework identifies a Casimir-friendly structure and provides a practical reduction that unifies constrained and unconstrained dynamics within a single formalism, with potential computational benefits for dissipative nonholonomic systems.

Abstract

In this paper we discuss nonholonomic contact Lagrangian and Hamiltonian systems, that is, systems with a kind of dissipation that are also subject to nonholonomic constraints. We introduce the so-called contact Eden bracket that allows us to obtain the evolution of any observable. Finally, we present a particular vector subspace of observables where the dynamics remain unconstrained.

Theorems & Definitions (10)

• Theorem 1
• Definition 1
• Theorem 2: segundo
• Lemma 3
• Theorem 4: segundo
• Corollary 1
• Theorem 5
• Proposition 1
• Theorem 6
• proof