Elementary theoretical description of the heavy symmetric top

Abstract

A simple theoretical description of a symmetric top with one stationary point and subject to the action of the gravitational force is presented without the use of Lagrangian mechanics. Also no use is made of the Euler angles. The fact that the vectors of the angular momentum l, the angular velocity ω, and the symmetry axis ẽ3 are coplanar every instant of time is used to derive a closed set of equations for ẽ3 and l. The frequencies of pure “slow” and “fast” precession are calculated without solving the differential equations and a simple interpretation of the latter is given. Using energy conservation, the well-known differential equation for the angle θ between the symmetry axis and the surface normal is derived. The combination of precession and nutation is discussed for small nutation amplitudes.