Here is the link to my paper that derives the complete mathematical
solution to the gyroscope (the spinning top) in closed form.
http://www.scribd.com/doc/95320068/Solving-the-Spinning-Top

The abstract is,

"The closed form solution is found for the fully nonlinear dynamics of the gyroscope with a fixed point at the tip. The solution is found by using Cardano’s formulae to factor a cubic, in the case where all roots are know to be real. From this, the nutation angle is solved first in terms of Jacobi’s elliptic integral of the first kind. A simple change of variables then transforms the dependent variable of the remaining equations from time to the projection of the nutation angle onto the vertical axis. After this transformation, the remaining equations can be integrated exactly, giving solutions expressed in Jacobi’s elliptic integrals of the third kind. Reduced energy, angular momentum and moment of inertia are defined."

The abstract is,

"The closed form solution is found for the fully nonlinear dynamics of the gyroscope with a fixed point at the tip. The solution is found by using Cardano’s formulae to factor a cubic, in the case where all roots are know to be real. From this, the nutation angle is solved first in terms of Jacobi’s elliptic integral of the first kind. A simple change of variables then transforms the dependent variable of the remaining equations from time to the projection of the nutation angle onto the vertical axis. After this transformation, the remaining equations can be integrated exactly, giving solutions expressed in Jacobi’s elliptic integrals of the third kind. Reduced energy, angular momentum and moment of inertia are defined."