Analysis of the steady motions of complex rotating systems

This paper develops a practical computational framework for analyzing the steady motions of complex rotating systems. Ignorable coordinates corresponding to the translational or rotational invariances of a closed system can always be used in order that the effective potential functional $\text{W}$ for the relative motion depends only upon the relative configuration. However, the stationary values (steady motions) of $\text{W}$ can be determined by an essentially unconstrained calculational procedure even when $\text{W}$ is expressed most simply with Cartesian coordinates. It is shown that the angular velocity of any fixed rotating configuration (FRC) must paralled a principal axis when some mild assumptions about the nonrigidity of the system are fulfilled. This property implies that every distinct FRC corresponds to a stationary value of $\text{W}$. The discussion includes examples of $\text{N-}\text{body}$ problems and a spin-stabilized kilometer-wave orbiting telescope. Random or systematic searching and a quasi-Newton algorithm are used to compute minima of $\text{W}$.