TAB 23

POINCARE SECTIONS

It is interesting to note that Poincare sections are a conformal mapping of concentric parallel slits in a circular domain. The key is to represent this as a bi-linear transformation, featuring an interplanetary trajectory. The problem is that the trajectory is going between two different fractal domains, which is illustrated quite nicely by this transformation because it is a combination of three separate linear transformations:

translation (f-force at the sun)

inversion (gravity)

stretching (g force at Jupiter)

The following illustration of the "system wave" shows the motion of the planets to exist, each in a unique "Poncare Section."

Poincare Sections on Phase Space

Closed noncircular periodic orbits can happen only with an inverse squared law force (i.e. gravity or electrostatic potential) or the harmonic oscillator potential which is a 1/r force ~ iff the ratio of the angular frequencies of the x- and y-motion is rational. In the mathematics these are called Lissajous orbits. (Note: the angular frequencies are equal for elliptical or straight line motion in the harmonic oscillator model.) "Commensurable" means the ratio is a rational fraction.

A typical Lissajous orbit, a closed periodic orbit, is shown in the next figure.

A Lissajous Orbit

Note that if the ratio of angular frequencies is different from a rational number by even an infinitesimal amount (i.e. both numbers must be known to infinite precision), then the motion is no longer closed and will fill up the square. Since pi and e are both irrational numbers, they cannot in any way be associated with a periodic orbit ~ thus confirming that circles are singularities. ( FIVE ) ) Which is odd, considering that for all bounded functions of two variables, the value at the center is equal to its average value on the circumference of the circle ~ e.g. nature does its utmost to achieve circular motion, despite its instability.

The Biharmonic Conformal Mapping

The above illustration shows the instability of a one proton/one electron system (the Bohr Atom), with the electron in a perfectly spherical orbit - e.g. no perturbations. A spherical charge distribution, as with gravity, induces a zero charge field within the sphere - a point singularity. This destabilizes the proton, causing motion; having no counterbalancing electric negative shell, the proton nucleus field begins to expand; bumping the electron out to a higher orbital; and it stabilizes in a higher order spherical s-type orbit; and so forth. Conformally, the two conditions are the same.

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© 2004 WH Clark