This constant " areal velocity " can be calculated as follows.
2.
The constancy of areal velocity may be illustrated by uniform circular and linear motion.
3.
See the article on areal velocity . d'Arcy also had an illustrious military career in the French army.
4.
Thus, the areal velocity is constant for a particle acted upon by any type of central force; this is Kepler's second law.
5.
In this case, the constant " k " of the spiral can be determined from ? and the areal velocity of the particle " h " by the formula
6.
The magnitude of " h " also equals twice the areal velocity, which is the rate at which area is being swept out by the particle relative to the center.
7.
Since the speed " v " is likewise unchanging, the areal velocity ?" vr " ?" is a constant of motion; the particle sweeps out equal areas in equal times.
8.
As it does so, the object's motion will be described by two vectors : a translation vector, and a rotation vector "'? "', which is an areal velocity vector : the Darboux vector.
9.
In Proposition 44 of his " Principia ", he showed that the difference is proportional to the inverse cube of the radius, specifically by the formula given above, which Newtons writes in terms of the two constant areal velocities, " h " 1 and " h " 2
10.
Conversely, if the motion under a conservative force "'F "'is planar and has constant areal velocity for all initial conditions of the radius "'r "'and velocity "'v "', then the azimuthal acceleration " a " ? is always zero.
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