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

More than 160 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the orientation and paths of moving objects undergoing three-axis rotations. Here it is shown that they provide a natural way of selecting an appropriate orthonormal 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 some 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 Constantin, Fefferman, and Majda and by Deng, Hou, and Yu. It is then shown how the quaternion formalism provides an alternative formulation in terms of the Hessian of the pressure.