Orbital classification of the integrable problems of Lagrange and Goryachev–Chaplygin by the methods of computer analysis

In an earlier paper Orel computed the trajectory (or orbital) invariant for the Goryachev–Chaplygin problem. She noted, however, that the investigation of this invariant is beset with insurmountable analytic difficulties. The present paper completes the construction of the trajectory invariant for the Goryachev–Chaplygin problem, as well as for the Lagrange problem, by the methods of computer analysis. Thus the orbital classification question as been solved for these problems. In the paper we also formulate conjectures relating to the Lagrange case, which can serve as a basis for further investigations in this area. The computational algorithms themselves are due to Takahashi.