| X = − | ∂V | , Y = − | ∂V | , Z = − | ∂V | . |
| ∂x | ∂y | ∂z |
(28)
The equation (24) or (26) now gives
1⁄2mv12 + V1 = 1⁄2mv02 + V0;
(29)
i.e. the sum of the kinetic and potential energies is constant when no work is done by extraneous forces. For example, if the field be that due to gravity we have V = ƒmg dy = mgy + const., if the axis of y be drawn vertically upwards; hence
1⁄2mv2 + mgy = const.
(30)
This applies to motion on a smooth curve, as well as to the free motion of a projectile; cf. (7), (14). Again, in the case of a force Kr towards O, where r denotes distance from O we have V = ∫ Kr dr = 1⁄2Kr2 + const., whence
1⁄2mv2 + 1⁄2Kr2 = const.