Stability and controllability in systems with quasicyclic coordinates

The theory of systems with quasicyclic coordinates is discussed with an application to a problem from satellite mechanics. The theory of systems with cyclic coordinates is extended to those with quasicyclic coordinates, particularly from the standpoint of the stability of simple movements. The definitions of cyclic and quasicyclic coordinates are reexamined and some new ones are suggested. The possibility of reduced systems with either type of coordinates, and of stable motions in such systems, is discussed. The conditions under which a system can be controlled, when the quasicyclic component of the generalized force can be used as the control variable, is addressed. A linear description of the controlled motion is obtained, and the phase space is constructed from relevant coordinates, their temporal derivations, and the quasicyclic velocities. The extent to which the rotation of a gyrostatic satellite can be controlled with a rotor is examined using a gyrostat model and a linear description of the motion.