Geometrical dissipation for dynamical systems

TL;DR

This paper introduces a geometric framework for constructing vector fields on Riemannian manifolds that conserve certain functions while dissipating others at a specified rate, with applications to physical models like the Landau-Lifschitz equation.

Contribution

It develops a geometric approach to dissipation in dynamical systems, linking gradient flows on regular leaves to physical dissipation models.

Findings

Vector fields of gradient type on regular leaves are characterized.

The framework unifies dissipation models such as Morrison dissipation.

Applications include reformulating the Landau-Lifschitz equation in a unitary form.

Abstract

On a Riemannian manifold $(M,g)$ we consider the $k+1$ functions $F_1,...,F_k,G$ and construct the vector fields that conserve $F_1,...,F_k$ and dissipate $G$ with a prescribed rate. We study the geometry of these vector fields and prove that they are of gradient type on regular leaves corresponding to $F_1,...,F_k$. By using these constructions we show that the cubic Morrison dissipation and the Landau-Lifschitz equation can be formulated in a unitary form.