In section 8.4 it was shown that the configuration set for the motion of a rigid rod was a 5-dimensional submanifold of R6and furthermore that the reaction (or “internal”) forces were orthogonal to this submanifold. Thus the “problem could be treated as one of a single particle moving on a 5-dimensional submanifold of R6.

In this chapter the motion of a rigid body moving in R3is considered and it is shown that the configuration space can be thought of as R3 × S0(3). Thus this problem reduces essentially to the motion of a single particle on a 6-dimensional configuration manifold. Furthermore the reaction forces are shown to be orthogonal to the configuration manifold. The derivation of Lagrange's equations given in Section 8.3. thus applies to rigid body motion when the “external” forces are due to a conservative field of force.

MOTION OF A LAMINA.

In section 8.4. we showed that Lagrange's equations were applicable to the problem of the motion of a rigid rod. There the “constraint condition” that the rod's length remained constant was sufficient to define the configuration manifold. By way of introduction to the problem of the rigid body we look at the motion on a plane of a “lamina” which we think of as being defined by three “point masses”. Its configuration set will clearly be a subset of R6 and we will again require that the distances between the particles remain constant. These three constraints, however, do not tell the whole story.