The dynamics of a symmetric top are exactly integrable, due to the existence of three integrals of motion: (1) angular momentum around the
; (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
where the angles
parametrize the position of the top's moving end point on the sphere,
parametrizes the top's rotation around its symmetry axis, 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
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:
 H. Goldstein, Classical Mechanics
, 2nd ed., Reading, MA: Addison Wesley, 1980.
 L. D. Landau and E. M. Lifshitz, Mechanics
, 3rd ed.: Course of Theoretical Physics
, Vol. 1, Oxford: Pergamon Press, 1976.