OK many thanks, I developed this in Note 373(3). In these notes the relativistic kinetic energy T = (gamma – 1) mc squared is expanded in a binomial series in (v / c) squared. In the definition of gamma, v is the Newtonian value.
Sent: 18/03/2017 16:26:47 GMT Standard Time
Subj: Re: 373(2): Analytical Orbit from the Relativistic Lagrangian
In (14) the radial trajectory r(t) of the non-relativistic case is used. A different v (here denoted v_1) changes r, so this can only be consered as one step in an iterative process. You express this by using a new radial variable r_1 which is correct. However The relation
v_1^2 = 2/r_1 – 1/a
only holds for Newtonian ellipses. This should be kept in mind. Parameters like epsilon and alpha are only defined exactly for non-precessing ellipses. But as an approximation I agree with your procedure.
Am 16.03.2017 um 15:41 schrieb EMyrone:
This is given in the first approximation to a binomial expansion as Eq. (17), which supplements the orbital precession found numerically by Horst Eckardt. The accuracy of the orbit (17) can be increased by adding more terms of the binomial expansion of the relativistic kinetic energy, Eq. (7). Experimentally, it is found that the perihelion advances by 3MG / (alpha c squared) every two pi radians. I accept the experimental claim for the sake of argument, but reject the Einsteinian general relativity. Dr. Horst Eckardt found by computer algebra that the relativistic potential is given by Eq. (20), which reduces in the limit v << c to the inverse square law, Eq. (21).