Many thanks indeed, very interesting results again! The x theory is developing strongly, these are entirely new results both in physics and mathematics. I will post them next on the blog.
In a message dated 26/06/2012 13:15:38 GMT Daylight Time, writes:
These are essentially the graphical results of paper 222.
plot %t15: The change of the radial component of the velocity is zero at the points of return.
plot %t20: the polar plot of the velocity shows the precession for x<>1.
plot %t25: the rate of area change dA/dr in polar representation is a straight line for x=1. For x<>1 several hyperbolas appear.
plot %t33/t34: the Area dependence A(theta) jumps when the argument x*theta crosses pi. This is an artefact of the definition of inverse trigonometric functions. At the jumps, the values have to be shifted to give a continuous line.
plot %t38: the integrated function A(r) is defined between minimal and maximal radius as is dr/dt. The slope depends on x.
plot %t42: the area A(r,x) has been shown for three fixed values of r. The behaviour is different for r near to r_max (decrasing instead of increasing function).
plot %t51: total area A for theta between 0 and 2 pi. This depends on x. There are jumps again due to the inverse trigonometric functions. Behaviour is similar to A(theta), %t33, although that is a quite different case.