High Frequency Behavior of a Rolling Ball and Simplification of the Separation Equation

The Chaplygin separation equation for a rolling axisymmetric ball has an algebraic expression for the effective potential $V (z = \cos\theta, D, \lambda)$ that is difficult to analyze. We simplify this expression for the potential and find a 2-parameter family for when the potential becomes a rational function of $z = \cos\theta$. Then this separation equation becomes similar to the separation equation for the heavy symmetric top. For nutational solutions of a rolling sphere, we study a high frequency $\omega_3$-dependence of the width of the nutational band, the depth of motion above $V (z_{min}, D, \lambda)$ and the $\omega_3$-dependence of nutational frequency $\frac{2\pi}{T}$.