Poinsot’s construction describes the motion of a freely rotating rigid body in terms of an ellipsoid rolling on a plane. (http://www.phys.ttu.edu/~huang24/Teaching/Phys5306/CH5C.pdf), and the path of the point of contact is called the herpolhode, and gives the direction of the angular velocity. It seems to me that this would give the best way of modelling the movement of a rigid body, rather than having to work around the problems with energy-changing errors (http://en.wikipedia.org/wiki/Precession, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.38.7744&rep=rep1&type=pdf). However, I've not been able to find anything which tells how to calculate the herpolhode. Does anyone know of any work on this, or is there a reason why it won't work?
I believe there is quite a large classical literature on the herpolhode. For example
and following pages. I actually came across the term first, I think, in Greenhill's book on the application of elliptic functions - seems to come up via the intersection of two quadrics.
Given that anything on rigid body motion must be a way of describing paths in the Euclidean group, I suppose a more accurate question would be: how does this as a way of talking about kinematics tally with more familiar charts (on SO(3), in particular)? Since the approach seems to have gone right out of fashion, it is presumably less convenient. One has to bear in mind that the old mathematical physics was quite largely devoted to closed-form solutions, so that redescriptions might work well for particular problems.
Edit: There is a treatment on pp. 152-155 of E. T. Whittaker's Treatise on Analytical Dynamics; the polar coordinates of the herpolhode come out in terms of standard Weierstrass elliptic functions (P, sigma and zeta).
The polhode and herpolhode have not completely gone out of fashion. In Michael Spivak's very recent new book "Physics for Mathematicians: Mechanics I" there is a very nice, discussion of Poinsot's geometric approach to rigid body motion on page 336, and then on pages 366--371 Spivak has added a section (Addendum 9B) titled "Secrets of the Herpolhode" where in his usual clear and nicely illustrated style he shows how to calculate the herpolhode.