Sergey Chaplygin

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.

This classical paper by S.A. Chaplygin presents a part of his research in nonholonomic mechanics. In this paper, Chaplygin suggests a general method for integration of the equations of motion for non-holonomic systems, which he himself called the "reducing-multiplier method". The method is illustrated on two concrete problems from non-holonomic mechanics.
This paper produced a considerable effect on the further development of the Russian nonholonomic community. With the help of Chaplygin’s reducing-multiplier theory the equations for quite a number of non-holonomic systems were solved (such systems are known as Chaplygin systems). First published about a hundred years ago, this work has not lost its scientific significance and is hoped to be estimated at its true worth by the English-speaking world.
This publication contributes to the series of RCD translations of Chaplygin’s scientific heritage. In 2002 we published two of his works (both cited in this one) in the special issue dedicated to non-holonomic mechanics (RCD, Vol. 7, no. 2). These translations along with translations of his other two papers on hydrodynamics (RCD, Vol. 12, nos. 1,2) aroused considerable interest and are broadly cited by modern researches.

This text presents an English translation of the significant paper [6] on vortex dynamics published by the outstanding Russian scientist S.A.Chaplygin, which seems to have almost escaped the attention of later investigators in this field. Although it was published more than a century ago, in our opinion it is still interesting and valuable.

This text presents an English translation of the significant paper [5] on vortex dynamics published by outstanding Russian scientist S.A. Chaplygin (1869-1942), which seem to have escaped the attention of later investigators in this field. Chaplygin's solution includes that of an elliptical patch of uniform vorticity in an exterior field of pure shear. Although it was published more than a century ago, in our opinion it is still interesting and valuable.