2006/10/31 by John Gibbon, J. D. Gibbon
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Applied mathematics #Euler equations #Euler's formula #Geometric Analysis and Curvature Flows #Geometry #Lagrangian #Mathematical analysis #Mathematics #Navier-Stokes equation solutions #Orthonormal basis #Physics #Pure mathematics #Quaternion #math-ph #math.MP #nlin.CD
paper · pdf · doi:10.1070/rm2007v062n03abeh004411
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
arxiv created 2007/03/09 · openalex publication_date 2007/06/30 · arxiv updated 2015/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.