Abstract
In this paper, the problem of motion of a rigid body about a fixed point under the action of a Newtonian force field is studied when the natural frequency value
w =
1
3
. This singularity appears in [1] and deals with different bodies being classified according to the moments of inertia. Using Poincaré's small parameter method [2], periodic solutions — with zero basic amplitudes — of the quasilinear autonomous system are obtained in the form of power series expansions, up to the third approximation, containing assumed small parameter. Also, Lagrange's gyroscope and Euler's one are derived as special cases from our solutions. At the end, the geometric interpretation of motion using Euler's angles is considered to show that the resulted motion is of regular precession type which depends on four arbitrary constants.