ar X iv : n lin / 0 60 70 20 v 1 [ nl in . C D ] 1 1 Ju l 2 00 6 Lagrangian particle paths & ortho - normal quaternion frames

New optical methods now allow experimentalists to track the trajectories of Lagrangian tracer particles in fluid flows at high Reynolds numbers. Independently, quaternions are used in the aerospace and computer graphics industries to track the paths of objects undergoing three-axis rotations. It is shown here that quaternions are a natural way of selecting an appropriate ortho-normal quaternion-frame (not the Frenet-frame) for a Lagrangian particle and of obtaining the equations for its dynamics. The method is applicable to a wide range of Lagrangian flows. Hamilton discovered the multiplication rule for quaternions on 16th October, 1843, as a composition rule for orienting his telescope, which had four cranks. This feature – that multiplication of quaternions represents compositions of rotations – has made them the technical foundation of modern inertial guidance systems in the aerospace industry where tracking the paths of moving rotating satellites and aircraft is of paramount importance (Kuipers 1999). The graphics community also uses them to control the orientation of tumbling objects in computer animations because they avoid the difficulties incurred at the north and south poles when Euler angles are used (Hanson 2006). Given the utility of quaternions in tracking the paths of rotating objects one might ask whether they would also be useful in tracking Lagrangian particles in fluid dynamical situations. Recently by using optical methods developed for tracking particles created in cosmic ray bursts, experiments in turbulent flows have developed to the stage where the trajectories of tracer particles can be detected at high Reynolds numbers (Voth et al. 2002); see Figure 1 in [La Porta et al. (2001)]. The usual practice in graphics problems is to consider the Frenet-frame of a trajectory which consists of the unit tangent vector, a normal and a bi-normal (Hanson 2006). In navigational language, this represents the corkscrew-like pitch, yaw and roll of the motion. While the Frenet-frame describes the path, it ignores the dynamics that generates the motion. Here we will discuss another ortho-normal frame associated with the motion of each Lagrangian particle, designated the quaternion-frame. Quaternion-frames may be envisioned as moving with the Lagrangian particles, but their evolution derives from the Eulerian equations of motion. Suppose w is a contravariant vector quantity attached to a tracer particle following the flow along characteristic paths dx/dt = u(x, t) of a velocity u.