The radiation reaction force, according to the Abraham-Lorentz theory at least, is proportional to the jerk; that is, it is third order in time derivative of the oscillator displacement. Since it is small, the motion is considered still to be periodic and the third order time derivative is reduced to a first order derivative times the square of the natural frequency. Consequently, the jerk is transformed into a radiation damping term leading a natural line breadth since the motion is no longer monochromatic. If R is the coefficient of resistence in the velocit term in the harmonic oscillator equation then 1/R is the time it takes to decrease the amplitude in the ratio e:1, which is the lifetime of the oscillator.
However, since the self-radiation force is proportional to the jerk, it will no longer have an inverse-square dependency on the radial distance from the source. To the best of my knowledge no one has ever asked what is the force dependency on the radial distance from the emitting source. We will determine that here.
We have determined the product of the radii of curvature of the support function, rho, and its polar reciproca, rho*,l to be
rho rho* = (r/p)^2 = (v/r d phi/dt)^3 = (a/v d phi/dt)^3 = (j/a d phi/dt)^3= (s/j d phi/dt)^3= ...(1)
where v is the velocity, a, the acceleration, j , the jerk, and s is the snap. The pedal, p, is related to the radial distance by p = r^(n+1), where n is a characteristic exponent, whose values give sinus spirals. If we introduce the inverse u=1/r, all the above equations reduce to
rho rho* = u^(3n), (2)
where we have neglected constant factors.
As an example, consider the first equality. Solving the pedal equation,
dp/dr = r/ rho.
gives rho= u^(n-1), excluding the case where n=-1, a straight line. Introducing this into (2)
gives the radius of the curvature of the polar reciprocal as
Newton's requirement that this be constant gives n=-1/2, which is a parabola. (Sinus spirals cannot account for conic sections where the eccentricity is different than 1. Nevertheless we can consider it as also representative of an ellipse.) From the expression of the product of the radii, it follows that v= rho d phi/dt= u^(n-1) d phi/dt, and dr/dt= u^n d phi/dt.
The acceleration is always given by
a= v^2/r = u^(2n-1) (d phi/dt)^2. (3)
The transmutation of the force, considers it how it changes when the source is displaced from a focus to the center of an ellipse. If the primes refer to the souce at the center of the ellipse, Chakerian in his "Central force laws, hodographs, and polar reciprocals" finds the accelerations to be given by
a = h^2 r/ rho p^3 and a' = h'^2 r'/ rho' p'^3. (4)
Considering a point P on the curve of the ellipse, he concludes that rho=rho' and h=h', where h is the angular momentum per unit mass. He thus comes out with
a/a'= (r/p^3)(p'^3/r'). (5)
Since p v =h, (4) can be written as
a/a' = (r v^3/h^3) (h'^3/v'^3 r')= (r/r') (v/v')^3.
This, however, contradicts (3), which we know to be correct. His conditions on rho and h imply
p/p' = v'/v = (d phi/dt)'/(d phi/dt),
which are overly restrictive. Rather, introducing the definitions of h = r^2 d phi/dt and rho = v/d phi/dt, we come out with
a/a' = (v^2/r) (r'/v'^2), (6)
which was to be expected.
We know by other means that a =1/r^2 at the focus and a'=r' at the center. These are Newton's and Hooke's laws, respectively. This gives the ratio of the velocites as
(v/v')^2 = 1/r r'^2,
which is the same as the ratio of accelerations with the prime and unprime being switched. Using Newton's law, we may verify our expression for the acceleration, (3). Equating it with u^2 we get
a = u^(2n-1) (d phi/dt)^2= u^2. (7)
Rearranging we get
(d phi/dt)^2 =u^(3-2n).
Only with n=-1/2, do we get the conservation of angular momentum, d phi/dt = u^2.
Alternatively, if we place ourselves at the center of the ellipse, (7) must be replaced by
a = u^(2n-1) (d phi/dt)^2= 1/u. (8)
Now, in order to get the conservation of angular momentum. we must set n=-2, which is a rectangular hyperbola, another sinus spiral. So for each conservation law we get a characteristic sinus spiral curve.
Now, turning our attention to the ratio of the jerks at a focus and the origin we get
j/j' = (a/a')^2 (v'/v) = 1/r' r^(7/2), (9)
since the jerk is j= a^2/v, analogous to the acceleration a= v^2/r and the snap, s = j^2/a. and crackle, c =s^2/j.
From (9) we see that j', like a', is proportional to r', but at the focus things are different. We now have
j = 1/r^(7/2), (10)
which has a much shorter range than Coulomb's 1/r^2 law. To verify (10) we set it equal to the definition of the jerk,
j = v^3/r^2 = u^(3n-1) (d phi/dt)^3 = u^(7/2). (11)
The last equality in (11) implies that
[u^(n-3/2) d phi/dt]^3 = const. (12)
This will be true for n=-1/2, the sinus spiral for a parabola, which also includes the ellipse.
Finally, in the case of snap, the ratio is
s/s' = (j/j')^3 (a'/a) = 1/r' r^5. (13)
Again s'~r' at the center, but
s = 1/r^5 (14)
at the focus. In regard to dual laws, (14) is a self-dual law consisting of a circle passing through the origin of the coordinates. Crackle has the ratio
c/c' =(s/s')^2 (j'/j) = 1/r' r^(13/2). (15)
While the force law at the center remains the same. the force law at the focus decreases as the 3/2 power as we go to higher time derivatives, e.g.
c = 1/r^(13/2).
Beyond crackle, we would come out with a 1/r^8 dependency.