Bi-Hamiltonian Structure of Gradient Systems in Three Dimensions and Geometry of Potential Surfaces

TL;DR

This paper demonstrates that all three-dimensional dynamical systems have compatible bi-Hamiltonian structures and explores how potential surface geometry influences Hamiltonian functions and conserved quantities.

Contribution

It establishes the existence of bi-Hamiltonian structures for all 3D systems and links potential surface geometry to Hamiltonian functions and conserved quantities.

Findings

All 3D dynamical systems possess two compatible Poisson structures.

Hamiltonian functions relate to distance functions on potential surfaces.

Application to Aristotelian three-body model reveals conserved quantities.

Abstract

Working bi-Hamiltonian structure and Jacobi identity in Frenet-Serret frame associated to a dynamical system, we proved that all dynamical systems in three dimensions possess two compatible Poisson structures. We investigate relations between geometry of surfaces defined by potential function of a gradient system and its bi-Hamiltonian structure. We show that it is possible to find Hamiltonian functions whose gradient flows have geodesic curvature zero on potential surfaces. Using this, we conclude that Hamiltonian functions are determined by distance functions on potential surfaces. We apply this technique to find conserved quantities of three dimensional gradient systems including the Aristotelian model of the three-body motion.