Many thanks again. The hamiltonian (1) is H = (gamma – 1) mc squared + U in which gamma = (1 – (v/c) squared) power minus half, where v is the Newtonian velocity v squared = r dot squared + r squared phi dot squared. So the Newtonian starting point (4) is self inconsistent, and needs to be replaced by r dot and phi dot from the numerical solution of the lagrangian. we know that the lagrangian solution leads to a precessing ellipse. When v << c the hamiltonian becomes H = (1/2 m v squared + U, and the lagrangian becomes L = (1/2) m v squared – U. I will think about this problem some more.
Sent: 18/03/2017 18:04:45 GMT Standard Time
Subj: Re: 373(3): Analytical Orbit from ECE2 Special Relativity
The formulas are correct so far. Since we used the parameter relations for a Newtonian ellipse, there is no precession in eq.(21). I plotted r(phi) and dr/dphi for three different values of c, representing a transition between a non-relativistic and relativistic orbit. r(phi) has a horizontal tangent at phi=0 and phi=2 pi, there is no precession. This can also be seen from the derivative plot. However the turning point of orbit r(phi) changes in the relativistic case, see extrema in th 2nd figure.
Am 18.03.2017 um 12:32 schrieb EMyrone:
The analytical orbit is Eq. (21), which can be checked by computer algebra and graphed. In Eq. (21) H sub 1 is a constant. The astronomically observed orbit is a precessing ellipse. By adjusting H1, exact agreement can be obtained with the astronomical data. This theory can be made more and more precise by considering more and more terms in the binomial expansion of Eq. (1). In immediately preceding UFT papers the orbit of the ECE2 lagrangian was found numerically to be a precessing ellipse.