Conservative dissipation: How important is the Jacobi identity in the dynamics?

TL;DR

This paper investigates the significance of the Jacobi identity in Hamiltonian and related conservative dynamics, showing its crucial role in shaping phase space structure through comparative analysis.

Contribution

It provides a comparative study of Hamiltonian, almost-Poisson, and metriplectic flows to highlight the importance of the Jacobi identity in dynamical systems.

Findings

Jacobi identity is essential for proper phase space structuring.

Differences observed between Hamiltonian and almost-Poisson flows.

Metriplectic flows show distinct dynamical properties.

Abstract

Hamiltonian dynamics are characterized by a function, called the Hamiltonian, and a Poisson bracket. The Hamiltonian is a conserved quantity due to the anti-symmetry of the Poisson bracket. The Poisson bracket satisfies the Jacobi identity which is usually more intricate and more complex to comprehend than the conservation of the Hamiltonian. Here we investigate the importance of the Jacobi identity in the dynamics by considering three different types of conservative flows in R3 : Hamiltonian, almost-Poisson and metriplectic. The comparison of their dynamics reveals the importance of the Jacobi identity in structuring the resulting phase space.