Cayley kinematics and the Cayley form of dynamic equations

The Cayley transform and the Cayley–transform kinematic relationships are an important and fascinating set of results that have relevance in N–dimensional orientations and rotations. In this paper these results are used in two significant ways. First, they are used in a new derivation of the matrix form of the generalized Euler equations of motion for N–dimensional rigid bodies. Second, they are used to intimately relate the motion of general mechanical systems to the motion of higher–dimensional rigid bodies. This approach can be used to describe an enormous variety of systems, one example being the representation of general motion of an N–dimensional body as pure rotations of an (N + 1)–dimensional body.