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Polar Reciprocals and Higher Order Physics

Updated: Sep 21, 2023

Regarding the significance of central force dual laws, Needham writes

Their physical significance [ integer power laws, -4,-5 and -7, in addition to +1 and -2] is unknown to me, and that in itself is strange, for the music of mathematics is seldom played with the accompanying echo being heard in Nature.

And that has turned out to be a loud echo! For a body acted upon by a central force varies inversely as the power 2n+3 of the distance. These forces produce trajectories that follow sinusoidal spirals, which itself is a misnomer since they are not spirals in general. The dual pairs are n=2 and n=1/2, corresponding to a lemniscate and cardioid, respectively. The case n=1 is an inverse fifth power law and corresponds to a circular trajectory going through the origin, the so-called pendulum orbits of the old quantum theory.

But, there is something even more deeper in all this since it involves polar reciprocals. The distance from the source to the tangent of the curve to which it is normal is known as the pedal, p The pedal to the radial distance to any point on the orbit is to the reciprocal of the pedal, r*=1/p as the reciprocal pedal, p* is to the radial distance, r, i.e. p*=1/r. The second curve found at the distance r* from the source such that r*p=1 is the polar inverse of the original curve. Nature has chosen it such that it has constant curvature.

Now the beauty of it all lies in the fact that it is not limited to the radial distance r and the pedal p. For if we substitute the velocity, v, for r and the angular velocity for the pedal, the polar reciprocal of the hodograph is obtained by rotating the latter by 90 degrees and rescaling by a factor of the inverse of the conserved angular momentum. This yields an expression for the acceleration which is Newton's inverse square law.

Proceeding to higher-order, we replace the velocity by acceleration and the angular velocity by the product of the velocity and the rate of change of the angular speed. This is orthogonal component to the tangent of the trajectory. But, only now, the trajectory occurs in velocity space and not in configuration space. The conservation law that makes this equivalent to configuration space is the conservation of angular momentum in velocity space: The velocity vector sweeps out equal areas in equal times--in velocity space.

Continuing, we replace the acceleration by the jerk, the third order derivative of the motion, and its normal component, the product of the acceleration times the rate of change of the angular speed. Surprisingly, the polar reciprocals form closed orbits, just like an ellipse under the inverse-square law. The beauty likes in the fact that this can be extended to snap, crackle and pop, which are still higher order derivatives in the motion. The trajectories now occur in acceleration space, and form closed orbits, just like the original polar reciprocals in configuration space. The radius of curvature becomes constant where the jerk varies inversely proportional to the inverse square of the velocity, Newton's law in velocity space!

In what follows, I shall develop each of these spaces and give the physical predictions upon which they can be tested. So mathematics does resonate in the physical world. It consists of isolating the pedal, or the distance between the source and the tangent to the curve for which it forms a right angle. These are always angular quantities, and the curves and their radii of curvature are consequences of the interplay between the total quantity and its normal component to the motion.

Analogous to dual force laws, there are polar reciprocals. If a point P traces out a pedal curve, a point P' determined OP x OP'=1 (i.e., inverses) traces out the polar reciprocal where the source is located at O. For instance, if P traces out a cardioid, the P' will trace out parabola. The former has index n=1/2, the latter has index n=-1/2. In contrast, for dual force laws, the cardioid, n=1/2, would be paired with the lemniscate, n=2. Inverse indices are replaced by pairs of indices of opposite sign.

Finally, we may look for a generalization of Newton's expression for the radius of curvature,

rho = v^3/w^3 / rho*

where w is the angular velocity and rho* is the radius of curvature of the polar inverse. It was Newton's great insight that led him to consider the radius of curvature of the polar reciproacal a constant. This leads to his inverse square law. The Newtonian slope r'/r, where the prime denotes the derivative with respect to the angle variable, can be generalized to the slope f"/f' for any continuous function f. The Schwarzian derivative measures how f changes the cross-ratio for infinitesimally close points. The cross-ratio is the measure of distance in non-Euclidean geometries. if z=r"/2r' generalizes the Newtonian slope then the Schwarzian is define as

(1/2) S= z'-z^2,

and its relation to the radius of curvature is


for the generalized definition of Newton's slope.

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