The Differential Equations of Motion

Abstract

To begin with we shall consider a free¹ system of mass points. We call it a system because we assume that the points are subject to external forces not independently of one another, in which case one could consider each point by itself, but as they act mutually on one another one cannot consider any one without considering the others. Further, the system is a free one, i.e., one in which the points follow the action of the forces unhindered. Let any point of the system have a mass m, and its rectangular coordinates at time t be x, y, z and the components of the force acting on it X, Y, Z; then one has the well-known equation of motion:

$$m\frac{{{d^2}x}} {{d{t^2}}} = X,\;m\frac{{{d^2}y}} {{d{t^2}}} = Y,\;m\frac{{{d^2}z}} {{d{t^2}}} = Z\;$$

, Similar equations hold for all points of the system. X, Y, Z depend on the coordinates of all n points and can also contain their derivatives with respect to time t, which is always the case when the resistance is to be taken into account.