# Herpolhode equation

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?

• There are some useful comments and references by Andrey Rekalo in response to an earlier MO question, Rolling a convex body: Geodesics vs. rolling curves, mathoverflow.net/questions/32163 . Mar 25, 2011 at 11:06

I believe there is quite a large classical literature on the herpolhode. For example

http://www.archive.org/stream/cu31924005727965#page/n477/mode/2up

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.

• Thank you for this post Dick ... our UW Quantum Systems Engineering (QSE) Group has just now ordered Spivak's book! Might you say whether quantum dynamics is discussed in it? (as yet, it's not on Google Books) E.g., Chapter 3 of Slichter's "Principles of Magnetic Resonance" (and indeed the rest of the book) is concerned with certain (quantum) conservation laws associated to spin transport. These laws are exact also on classical state-spaces and (amazingly to us) even on arbitrary-rank tensor-network state-spaces. We wish we understood this better, and we hope that Spivak may explain it! Mar 25, 2011 at 17:33
• @John Sidles Sorry, no, this first volume treats only classical mechanics from its first page to the last. Mike plans eventually to have a volume on QM, but that will probably be well in the future. (He told me that E & M is next on his list.) Dick Palais Mar 25, 2011 at 23:48