Beautiful, Interesting Quaternions Are Valuable for Rigid Bodies
March 2003 page 12
The picture-caption story in the October 2002 issue of Physics Today (page 23) stated that Murray Gell-Mann would be delivering the Royal Irish Academy's inaugural Hamilton Lecture at Trinity College Dublin this year. Gell-Mann was quoted as saying, "They are celebrating Hamilton's quaternions, which are beautiful and mathematically interesting, even though they never proved to be of that much use for physics." Classical mechanics may be now relegated to applications, and may not be regarded as useful to physics. However, quaternions are useful in the treatment of the rigid-body problem. The formulation was achieved by the late Harold S. Morton Jr.1
The state of the body is expressed in terms of the four Euler parameters and their four canonically conjugate momenta. The Euler parameters are the elements of a quaternion, subject to the constraint that the norm, the sum of the squares of the elements, is unity. That constraint is essential in the formulation.
Morton includes a numerical example for a torque-free rigid body. I wrote a Fortran code to implement these equations of motion for the case of a spinning undeployed spacecraft during the ascent phase of its motion, between separation from the launch vehicle and the application of a final maneuver that placed the spacecraft in a near-mission orbit. The results were in complete agreement with established alternative models.
The great advantage of Morton's formulation is that the quaternion elements are mathematically very well behaved and are not subject to singularities, such as those encountered with Euler angles when the directions about which the angles are measured nearly coincide.
The quaternion method can easily be made part of the treatment of a many-body problem, such as that of a spacecraft containing spinning angular momentum wheels, which are often operating during ascent.
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