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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.
 

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
vector distributions.
DOI: 10.1134/S0081543807030030
1. INTRODUCTION
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.

  

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

 

Collections: Engineering; Mathematics