Feed: Dr. Myron Evans
Posted on: Monday, May 13, 2013 11:51 PM
Author: metric345
Subject: Checking the New Orbital Equation by Computer
We seem to have discovered a very rich new area of orbital theory. The computer is a great help, especially when controlled in the incisive way demonstrated here by co author Dr. Horst Eckardt. The new equation is a novel combination of the two orbital Lagrangian equations, which have been well known since the late eighteenth century. It is also interesting that the diffusion equation has a structure similar to the diffusion equations of Brownian motion theory such as the Fick, Smoluchowski and Kramers equations which are known to have a very rich mathematical structure and many classes of solution. These are also similar to the Schroedinger equation structure and solutions. In this theory, the precession of the perihelion is computed exactly to machine precision, so the near circular approximation is not needed. For the Newtonian potential there should be no precession of the ellipse, but for other force laws, there should be a precession. Using this method r can be computed as a function of t and t as a function of r and theta. So this method can be used for animation. It can also be applied to galaxies and used with any force law. The heavy calculational work is all done by the computer. This approach would be impossible by hand because of all the calculations.
To: EMyrone@aol.com
Sent: 13/05/2013 17:32:31 GMT Daylight Time
Subj: Re: 242(3): Equation for Theta for Any Planar Orbit and Any Force Law
I checked the analytical solution. First the original harmonic oscillator gives a real solution, but this does not matter.
In the second section the general solution is developed.
In the third section the equation is rewritten with F(r) instead of Omega.
The fourth section is a check with both constants = 0 and a constant Omega. The complex solution
r = exp(i omega t)
appears. Obviously the first constant is needed to allow for a radius scaling required for correct physical dimensions.
Setting
%k1 = – 1/2 * v0
(this is a velocity) leads to the (real) result
r = v0/omega * sin(omega t)
so we have
r0 = v0/omega.
I used the second solution, anyhow the sign seems to play a role. It is also important that %k1 is negative, otherwise a complex radius comes out.
Horst.
242(3).pdf

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