AN INTEGRABLE CASE OF A ROTATIONAL MOTION ANALOGOUS TO THAT OF LAGRANGE AND POISSON FOR A GYROSTAT IN A NEWTONIAN FORCE-FIELD

The scope of the present paper is to provide analytic solutions to the problem of the attitude evolution of a symmetric gyrostat about a fixed point in a central Newtonian force field when the potential function is V(2). We assume that the center of mass and the gyrostatic moment are on the axis of symmetry and that the initial conditions are the following: psi(t0) = psi0, theta(t0) = theta0, phi(t0) = phi0, omega1(t0) = 0, omega2(t0) = 0 and omega3(t0) = omega3(0). The problem is integrated when the third component of the total angular momentum is different from zero (B1 not-equal 0). There now appear equilibrium solutions that did not exist in the case B1 = 0, which can be determined in function of the value of l3r (the third component of the gyrostatic momentum). The possible types of solutions (elliptic, trigonometric, stationary) depend upon the nature of the roots of the function g(u). The solutions for Euler angles are given in terms of functions of the time t. If we cancel the third component of the gyrostatic momentum (l3r = 0), the obtained solutions are valid for rigid bodies.