Precession of the Kovalevskaya and Goryachev — Chaplygin Tops

The change of the precession angle is studied analytically and numerically for two classical integrable tops: the Kovalevskaya top and the Goryachev — Chaplygin top. Based on the known results on the topology of Liouville foliations for these systems, we find initial conditions for which the average change of the precession angle is zero or can be estimated asymptotically. Some more difficult cases are studied numerically. In particular, we show that the average change of the precession angle for the Kovalevskaya top can be non-zero even in the case of zero area integral.