Maxwell, in his Matter and Motion, writes
The study of the hodograph, as a method of investigating the motion of a body, was introduced by Sir W. R. Hamilton. The hodograph may be defined as the path traced out by the extremity of a vector which continually represents, in direction and magnitude, the velocity of a moving body.
Bohlin, Kasner, and Arnol'd have singled out integral powers laws with negative exponents 4, 5, and 7, in addition to the pair -1 and 2, as being paired through complex mappings. The latter are paired by the squaring a complex variable and correspond to the harmonic oscillator and inverse-square laws. This was associated with the displacement of the source at the center of an ellipse to one of its foci. However, the transformation also alters the nature of the orbit so it is not simply a matter of transporting the sun from the center to a focus of the same ellipse. In the words of Needham, "the music of mathematics is seldom played without an accompanying echo being heard in Nature." So what do these "magic" numbers refer to?
We want to show that the path of a particle in velocity space moving with respect to an integral power law force is a sinus spiral. If the sinus spiral is closed, its inverse will be open, and those magic numbers correspond to closed path sinus spirals.
The Binet equation, we derived in the last blog, can be written generally as
w"+w=c^(m-1)w^m, (*)
where w=1/v, the inverse velocity, and c is a velocity amplitude. This equation is derived from the transverse component of the jerk with a time rate of change of force given by
dF/dt=L^2c^(m-1)w^(m+2), (**)
where L is the conserved angular momentum in velocity space. This is derived from the vanishing of the normal component of the jerk.
The solutions to (*) are sinus spirals
w=(1/c)(cos(n(theta)))^1/n. (***)
Substituting (***) into (*) gives the relation
2n=1-m. (iv)
We are interested in integer m since these will correspond to integral power laws for (**). The dual laws with negative exponents 4, 5, and 7 correspond to m= 2, 3, and 5, respectively.
Now, for m=2, (iv) gives n=-1/2. The sinus spiral is a parabola in inverse-velocity space, corresponding to a cardioid in velocity space. This was what was obtained by solving Newton's law for the inverse-square law in configuration space. It corresponds to an inverse-4th law in velocity space.
The one we have derived previously corresponds to m=3, an inverse-5th law. In this case, n=-1, which is a straight line in inverse-velocity space, corresponding to a circle passing through the origin in velocity space. This is the limiting case where the eccentricity is equal to one.
The choice, m=5, corresponds to n=-2 which is an orthogonal hyperbolic in w-space. This corresponds to a lemniscate in v-space, that is governed by an inverse-7th power law.
For negative values of m, specifically m=-3, n=2 which corresponds to a lemniscate in w-space. In v-space it is an orthogonal hyperbola. The open orbit is produced by a Hookean law for the time-rate-of-change of the force in v-space.
Another negative value of interest is m=-1. This sinus spiral is a circle in w-space since n=1, while a straight line in v-space governed by an inverse-velocity law. So negative values of m are to associated with open orbits in v-space.
The borderline value m=0, corresponding to n=1/2 is a cardioid in w-space and a parabola in v-space. This corresponds to the inverse-square law for the rate of change of the force.
What would the velocity-dependent force look like? Consider
dF/dt=(Lc)^2/v^5
which produces a circular orbit through the origin. Transforming to the angle variable,
dF/d(theta)=Lc^2/v^3=c^2/p,
where we used the conserved angular momentum L=v^3/p=v^2d(theta)/dt, where p=v/d(theta)/dt is the radius of curvature. Integrating, and using the definition of the radius of curvature,
p=r/cos^3(theta){1+2r'^2/r^2/r^2-r"/r},
we get
F=c^2 int {cos^3(theta)(u+u") d(theta),
where u=1/r. Now, suppose that the inverse-square law holds in configuration space so that
u"+u=GM/h^2,
which uses the conservation of angular momentum, h, in configuration (c-)space. The integral can easily be performed and the result is
F=(GM c^2/h^2)(sin(theta)-sin^3(theta)/3), (v)
having set the integration constant equal to zero. The terms in (v) can be easily reduced to sinus spirals giving as a final result
F(v)=(3/4)(GM c^2/h^2)(c/v-c^3/v^3), (vi)
a combination of inverse-power velocity laws for the central force. Notice that all vestiges of the conservation of angular momentum in v-space has (conveniently) disappeared!
The Riemann metric of constant curvature can be obtained by setting (vi) equal to dv/dt, according to Newton's second law. Introducing the arc length, via dt=ds/v, and transforming to inverse v-space, w=1/v, we get the v-space metric
ds^2 = C(w) dw^2/(1-w^2c^2)^2,
where C(w) is a conformal factor proportional to 1/w^8. It is important to appreciate that this is a hyperbolic metric of constant curvature, 1/c. This is in contrast with the usual procedure that uses Levi-Civita time renormalization, and the v-space metric is elliptic.
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