Again, the velocities parallel and perpendicular to OP change in the time δt from u, v to u − v δθ, v + u δθ, ultimately. The component accelerations at P in these directions are therefore
| du | − v | dθ | = | d2r | − r ( | dθ | )2, |
| dt | dt | dt2 | dt |
| dv | + u | dθ | = | 1 | d | ( r2 | dθ | ), | |
| dt | dt | r | dt | dt |
(14)
respectively.
In the case of a central force, with O as pole, the transverse acceleration vanishes, so that
r2 dθ / dt = h,
(15)
where h is constant; this shows (again) that the radius vector sweeps over equal areas in equal times. The radial resolution gives
| d2r | − r ( | dθ | )2 = −P, |
| dt2 | dt |