Summary: ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2007, Vol. 258, pp. 1322. c Pleiades Publishing, Ltd., 2007.
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 258, pp. 1727.
Rolling Balls and Octonions
A. A. Agracheva,b
Received November 2006
Abstract--In this semi-expository paper we disclose hidden symmetries of a classical nonholo-
nomic kinematic model and try to explain the geometric meaning of the basic invariants of
The paper is dedicated to Vladimir Igorevich Arnold on the occasion of his 70th birthday. This
is just a small mathematical souvenir, but I hope that Vladimir Igorevich will get some pleasure
looking it over. The content of the paper is well described by the cryptogram below. Figure 1
represents the root system of the exceptional Lie group G2 (the automorphism group of octonions)
and two circles touching each other whose diameters are in the ratio 3 : 1.
Our starting point is a classical nonholonomic kinematic system that is rather important in
robotics: a rigid body rolling over a surface without slipping or twisting. The surface is supposed
to be the surface of another rigid body, so that the situation is, in fact, symmetric: one body is
rolling over another. We also assume that the surfaces of the bodies are smooth and cannot touch
each other at more than one point.