Alpha Institute for Advanced Studies

In a message dated 28/04/2012 13:15:22 GMT Daylight Time, writes:

The background that p/q must be rational to give closed orbits is probably the following:

An orbit closes if

r_1(theta+ m*2 pi) = r_1(theta + n*2 pi)

for a point of return r_1 and two integers m and n. If we describe the advancement of theta for one round by

theta –> x*theta,

this means there must exist an x with

x = n/m,

i.e. x must be rational.

Horst

Am 27.04.2012 15:56, schrieb EMyrone

Many thanks again, very important results. All of them could be analysed in UFT216 Section 4 and subsequent papers.

In a message dated 27/04/2012 14:01:48 GMT Daylight Time,

Prof. Evans,

I made my own plots for the parabola and variation of x. I see the same relationship holds as for the ellipse in that one must express x as a fraction p/q in reduced terms. The parabola will repeat p times and q will determine the closed areas which a bisector line will intersect. Also, an irrational value of x will eventually fill the plane if one keeps plotting more and more parabolas.

The plots attached have both a near and far views. One can confirm the equation for asymptotes occur at 2Pi(x-1)/x or in terms of deflection angle of a straight path as pi(2-x)/x.

Also, I noticed a ‘repulsive’ type notch right at the perihelion of some. I used Maxima to plot 3D curvature (z-axis) vs x and vs theta. I do see between x = 1.4 to x = 1.6 that the curvature goes negative for some range of theta. (Let me know if you can’t view an ‘EMF’ format graphic file – this was what Maxima/Gnuplot exports it as.)

Ray D.