New integrable problems in the dynamics of rigid bodies with the kovalevskaya configuration. I — The case of axisymmetric forces

Abstract

Despite the rarity of integrable problems in rigid body dynamics, amazing relative abundance of these problems is observed when the body has the famous Kovalevskaya configuration A=B=2C. In the present work, the general problem of motion of such a rigid body about a fixed point under the action of axially symmetric conservative forces is considered. The classical problem of motion of a heavy rigid body or a gyrostat and the problem of motion of a body in an ideal fluid are special cases. Three new integrable cases valid for Kovalevskaya's configuration are introduced of which one is general and the other two are restricted to the zero level of the cyclic integral. It is also shown that all the previously known results concerning the preesent problem are special versions of four different cases.