A Study of Relative Motion in Connection with Classical Mechanics

Abstract

Derivation of equations of motion of the second order, containing relative coordinates only.---This article is a continuation of the work of many authors, among others Berkeley, Neumann, Maxwell, Lange, Mach, Boltzmann, F\"oppl, and Kolkmeyer, whose work is briefly discussed. The main difficulty in Newton's mechanics seems to be to define the system of axes (inertial system) to which the motion is referred. Three solutions have been proposed. F\"oppl defined the system on a relative basis. Another and apparently more direct solution is to refer the motion of particles to other particles, that is, to use "relative coordinates" in the equations of motion. A third solution would be to obtain invariant equations, the same for all systems of reference. Introducing an hypothesis stated by F\"oppl, that for a system with its origin at the center of mass the angular momentum of the universe vanishes, invariant equations are obtained from the Newtonian, (though this is sometimes denied to be possible), and they degenerate into the Newtonian form if certain functions of coordinates and velocities occurring in them become zero by a suitable choice of the system of reference. It is believed, however, that such equations do not offer a complete solution of the problem of relativation of motion so as to satisfy the physicist; whereas the introduction of relative coordinates does. In this article this second solution has been applied to the Newtonian theory. Equations of motion are derived by introducing F\"oppl's hypothesis, which are of the second order and contain relative coordinates only. These equations may be taken as the basis for a complete system of mechanics. As examples the systems of Ptolemy and of Kepler are worked out.Application of relative coordinates to Einstein's theory of gravitation.---It is believed that the possibility of introducing relative coordinates is not limited to the Newtonian theory. As an example of the application of these coordinates to Einstein's theories, the case of the motion of a heavy and a light particle is considered and it is suggested that an hypothesis similar to F\"oppl's may give equations of motion into which only the distance apart of the particles measured by an observer on the heavier, enters.