Bi-Hamiltonian Structure in Serret-Frenet Frame

TL;DR

This paper presents a method to construct bi-Hamiltonian structures in three dimensions by solving a Riccati equation within the Serret-Frenet frame, linking solutions to compatible Poisson structures.

Contribution

It introduces a novel approach to derive bi-Hamiltonian structures using Riccati equations in the Serret-Frenet frame, connecting solutions to Poisson structures.

Findings

Bi-Hamiltonian structures can be constructed via Riccati equations in the Serret-Frenet frame.

Solutions of the Riccati equation lead to compatible Poisson structures.

Normal vectors of the moving frame also imply compatible Poisson structures.

Abstract

We reduced the problem of constructing bi-Hamiltonian structure in three dimensions to the solution of a Riccati equation in moving coordinates of Serret-Frenet frame. We then show that either the linearly independent solutions of the corresponding second order equation or the normal vectors of the moving frame imply two compatible Poisson structures.