Areal velocity and angular momentum for non-planar problems in particle mechanics

Abstract

The concept of areal velocity is intrinsically and historically connected with that of angular momentum. For central force fields the area swept out in the orbital plane by a particle’s radius vector is proportional to the time. For non-planar problems, it is shown that a particle’s position vector sweeps out a general conical surface and that interesting kinematical relations hold. The areal velocity vector is always perpendicular to the conical surface and is proportional to the angular momentum of the particle. In some problems, the magnitude of the areal velocity is constant while its direction changes. In others, only a component of the areal velocity vector is constant. A moving orthonormal basis associated with the areal velocity is defined and a matrix equation for the angular velocities of the basis vectors is found to have an elegant form, involving “sweeping” and “tilting” components. Illustrative examples are provided and some historical background is included.