Rigid Body Dynamics using Equimomental Systems of Point-Masses

The inertia matrix of any rigid body is the same as the inertia matrix of some system of four point-masses. In this work the possible disposition of these point-masses is investigated. It is found that every system of possible point-masses with the same inertia matrix can be parameterised by the elements of the orthogonal group in four dimensions modulo permutation of the points. It is shown that given a fixed inertia matrix, it is possible to find a system of point-masses with the same inertia matrix but where one of the points is located at some arbitrary point. It is also possible to place two point-masses on an arbitrary line or three of the points on an arbitrary plane. The possibility of placing some of the point- masses at infinity is also investigated. Applications of these ideas to rigid-body dynamics is considered. The equation of motion for a rigid body is derived in terms of a system of four point-masses. These turn out to be very simple when written in a 6-vector notation.