Ortho-normal quaternion frames, Lagrangian evolution equations and the three-dimensional Euler equations

TL;DR

This paper introduces quaternion-based methods for analyzing the dynamics of particles in three-dimensional flows and explores their implications for understanding potential singularities in the Euler equations.

Contribution

It develops a quaternion-frame approach for Lagrangian flow analysis and applies it to the Euler equations, offering new insights into vorticity and singularity formation.

Findings

Quaternion frames naturally describe particle rotation in flows

The quaternion approach relates to vorticity direction and pressure Hessian

Implications for finite-time singularity in Euler equations

Abstract

More than 150 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the paths of moving objects undergoing three-axis rotations. It is shown here that they provide a natural way of selecting an appropriate ortho-normal frame -- designated the quaternion-frame -- for a particle in a Lagrangian flow, and of obtaining the equations for its dynamics. How these ideas can be applied to the three-dimensional Euler fluid equations is then considered. This work has a bearing on the issue of whether the Euler equations develop a singularity in a finite time. Some of the literature on this topic is reviewed, which includes both the Beale-Kato-Majda theorem and associated work on the direction of vorticity by both Constantin, Fefferman & Majda and Deng, Hou and Yu. It is then shown how the quaternion formulation provides a further direction of vorticity result using the Hessian of the pressure.