On the Existence of Periodic Solutions of a Gyrostat Similar to Lagrange’s Gyroscope

Abstract

In this paper, the problem of the existence ofperiodic solutions of motion ofa gyrostatfixed at one point under
the action ofa central Newtonian force field, and a gyrostatic momentum l_i (i : 1,2,3; l₁ = l₂ = 0, l₃ not equal 0)
similar to a Lagrange gyroscope is investigated. We assume that the center of mass G of this gyrostat is
displaced by a small quantity relative to the axis of symmetry, and that quantity is used to obtain the small
parameter 8 (Elfimov, 1978). The equations of motion will be studied under certain initial conditions ofmotion.
The Poincaré small parameter method (Malkin, 1959; Nayfeh, 1973) is applied to obtain the periodic solutions
of motion. The periodic solutions for the case of irrational frequencies ratio are given. The periodic solutions
are geometrically interpreted to give the forms of Euler angles.