On Some Generalization of the Area Theorem with Applications to the Problem of Rolling Balls

This publication contributes to the series of RCD translations of Sergey Alexeevich Chaplygin’s scientific heritage. Earlier we published three of his papers on non-holonomic dynamics (vol. 7, no. 2; vol. 13, no. 4) and two papers on hydrodynamics (vol. 12, nos. 1, 2). The present paper deals with mechanical systems that consist of several spheres and discusses generalized conditions for the existence of integrals of motion (linear in velocities) in such systems.
First published in 1897 and awarded by the Gold Medal of Russian Academy of Sciences, this work has not lost its scientific significance and relevance. (In particular, its principal ideas are further developed and extended in the recent article "Two Non-holonomic Integrable Problems, Tracing Back to Chaplygin", published in this issue, see p. 191). Note that non-holonomic models for rolling motion of spherical shells, including the case where the shells contain intricate mechanisms inside, are currently of particular interest in the context of their application in the design of ball-shaped mobile robots.
We hope that this classical work will be estimated at its true worth by the English-speaking world.