In 1873 Bertrand stated, and supposedly proved, a theorem to the effect that among the central force potentials with bound orbits there are only two types of central force (radial) scalar potentials with the property that all bound orbits are also closed orbits.

This elevates both Hooke's and Newton's law to a dominating role in cosmology. He did so by considering the perturbed equation of an orbit taking as the fundamental solution a cos(n theta), where a is the amplitude and n an integer.

However, there is another theorem stating that the path of a particle moving according to a

central force is proportional to an (inverse) power of the radial coordinate is a sinus spiral,

a[cos(n theta)]^(1/n). This is an exact solution of the equation of the orbit!

A sinus spiral is a closed curve given by

r^n=a^n cos(n theta).

Bertrand's mistake was to consider r=a cos(n theta) as the valid solution for small perturbations. It is not a sinus spiral closed curve except in the case n=1, a circle. And,

as a matter of fact, that solution has a central force which is the inverse fifth other of the radial coordinate, thereby violating Bertrand's own theorem!

Bertrand's theorem gives the right-hand side of the equation of the orbit in terms of u=1/r as

J(u)~u^(1-b^2)

where the exponent b was found have values 1 (inverse square) 2 (harmonic motion) and 0 (circular motion), based on perturbation theory.

Yet, we know the exact expression for J(u) which is

J(u)~ u^(2n+1)

so that upon comparing with Bertrand's expression, we find

2n=-b^2

or n is always negative. But we know that there exist positive values of n, like n=2 for the lemniscate, so that Bertrand's proof is invalidated.

The central force law is

F=(n+1)a h^2/r^(2n+3).

For n=-1/2 and n=-2, we get Newton's and Hooke's law, respectively. But there are a whole host of central force laws, some of which are the dual laws. For n=2, the sinus spiral is a parabola, while for n=-2 it is an orthogonal hyperbola. The real part of the parabola consists of a Lissajous ellipse with its center of attraction at the center, and not at a focus with orbits bound above and below by the semi-major and semi-minor axes, respectively. But, this is none other than what Hooke's law claims. So Newton's and Hooke's laws are united by the sinus curve being a parabola.

Hooke's law corresponds to the orthogonal hyperbola, which by applying the inverse transform becomes a lemniscate (figure 8). This brings the points at infinity of the hyperbola

together at the origin of the lemniscate. The lemniscate which resembles two Roche lobes of a binary star has a central force which is an inverse seventh power of the radius, and is one of the trinity of dual laws. It is coupled to the inverse fourth power of the radius which is a cardioid. The lemniscate is a particular case of a Cassini oval, which is obtained by a perturbation of the inverse-seventh law, in exactly the same what that the cardioid is a particular case of a Descarte oval, whose force law is obtained by a perturbation of the the inverse fourth law.

In this way there is a complete analogy with intermolecular potentials and interstellar ones. In fact the latter makes a state about the former. In the Lennard-Jones 6-12 potential, corresponding to a Cassini oval, the 6 comes from the lemniscate, and the 12 should be replaced by 10, the short-range repulsive force. The only reason an inverse twelfth force was chosen was it was the square of the inverse-sixth, and only made calculations easier. There was absolutely no theoretical justification for it.

Referring to the two other pairs of the trinity of dual power laws, Needham states that "their physical significance is unknown to me, and that in itself is strange, for the music of mathematics is seldom played without an accompanying echo being heard in Nature."

The echo has been heard, and the epicycles that Newton's revolution did away with have made a come-back.

## Comentários