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

22 pages. This review is based on an invited lecture given at the meeting Mathematical Hydrodynamics held at the Steklov Institute Moscow, in June 2006

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• Particles and fields in fluid turbulence
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• Measurement of Lagrangian velocity in fully developed turbulence
• Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations
• The 3D structure of the Lagrangian acceleration in turbulent flows
• Quaternions and particle dynamics in the Euler fluid equations
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• Nature of complex singularities for the 2D Euler equation
• Lagrangian particle paths and ortho-normal quaternion frames
• Computational Euler History
• Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics

Cited by in corpus (4)

• Incompressible Euler Equations: the blow-up problem and related results
• On blow-up "twistors" for the Navier--Stokes equations in $R^3$: a view from reaction-diffusion theory
• Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics
• 3D Euler equations and ideal MHD mapped to regular systems: probing the finite-time blowup hypothesis