Phase flow of an axially symmetrical gyrostat with one constant rotor

Abstract

We analyze the attitude dynamics of an axially symmetric gyrostat under no external forces and one constant internal spin. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. With these coordinates, we realize that the problem belongs to a general class of Hamiltonian systems, namely the problem here considered is the one parameter Hamiltonian that is a polynomial of at most degree two in a base of the Lie algebra so (3). The parametric bifurcations are found for both cases, when the rotor is spinning about the axis of symmetry of the gyrostat, and when it is spinning about another axis of inertia. The general solution for the global general flow is expressed in terms of the Jacobian elliptic functions.