Nutation of a Symmetric Top

The dynamics of a symmetric top are exactly integrable, due to the existence of three integrals of motion: (1) angular momentum around the axis ; (2) angular momentum around the top's symmetry axis ; (3) energy of the system .

The equations of motion governing the dynamics of the symmetric top are

,

, and

,

where the angles and parametrize the position of the top's moving end point on the sphere, parametrizes the top's rotation around its symmetry axis, and and are the top's main moments of inertia with respect to the top's fixed point. The energy conservation restricts to angles between two extremal values, corresponding to two roots between -1 and 1.

Nutation of smooth type occurs when the sign of stays unchanged as varies between the two extrema. Nutation of cusp type occurs when reaches zero but does not change sign, and nutation of loop type occurs when changes sign.

The plot on the right shows the third-order polynomial in resulting from conservation of energy. The red line corresponds to .

Further details on the dynamics of a symmetric top can be found in the following textbooks:

[1] H. Goldstein, Classical Mechanics, 2nd ed., Reading, MA: Addison Wesley, 1980.
[2] L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed.: Course of Theoretical Physics, Vol. 1, Oxford: Pergamon Press, 1976.