The system is depicted by the following figure.
Characteristic fixed parameters are:
- $m$ for mass;
- $g$ for gravity constant;
- $\ell$ is the distance between the contact point and the center of mass;
- angles $(\phi,\theta,\psi)$ are for precession, nutation and rotation of the top.
- inertias $I_1=I_2\neq I_3$
The system Lagrangian is usually given as $$ L = \frac12 I_1 \left(\dot\theta^2+\dot\phi^2\sin^2\theta\right) + \frac12 I_3 \left(\dot\psi + \dot\phi\cos\theta\right)^2 $$
Evolution equations follow from applying Euler-Lagrange equations.
I am looking for references that contain and explain how to obtain exact or closed-form solutions to the spinning heavy symmetric top motion with a fixed contact point.
Particularly, I am looking for solutions that apply for simulations that are longer than 1 second.
I have consulted some "classical" references that treat the subject:
- Goldstein: Classical Mechanics (explains the physics well but does not detail a procedure to find analytical solutions)
- Whittaker: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (best one so far; provides a procedure to find analytical solutions but only for $0\leq t \leq 1$)
- Arnold: Mathematical methods of classical mechanics by Arnold (very similar to Goldstein's book in this matter)
Is there some reference that treats this problem and provides a procedure for finding analytical solutions that extend for $t\geq 1$?
Edit
There is also the classic (and large) book from Klein and Sommerfeld The Theory of the Top (in 2 volumes). However, this reference, although it describes most of what is to be known about the system, does not provide analytical solutions for $t\geq 1$.
