Many thanks again! Very fine results.The difference in Figure (9) is related to the experimentally observed precession of the perihelion, so to finish UFT373 I will attempt to find a relation between delta (dphi / dr) and the astronomically observed precession angle, 3MG / (c squared alpha).
In a message dated 21/03/2017 14:26:07 GMT Standard Time writes:
According to note 6, I first resolved eq.(3) after v^2. This gives a highly complicated equation with 4 solutions. Two solutions are complex, one is v=c and the fourth is real-valued. I used the fourth solution and inserted eq.(5). Then a highly complicated expression for (dr/dphi)^2 follows. Unfortunately it is complex. The real and imaginary part are plotted in Fig. 8. The result depends on the choice of constant H0. With
H= – 3.75 and H0= – 3.70
the realpart in Fig. 8 starts to become positive at the minimum radius of the ellipse. In so far this end behaves correctly, there is no zero crossing of (dr/dphi)^2 at the other end (r=2).
Alternatively I did the following: I resolved both Hamiltonians
H = 1/2 m v^2 + U(r)
H0 = (gamma-1) m c^2 + U(r)
separately according to v^2. From H follows the non-relativistic form eq.(5). Equating both solutions for v^2 gives an equation for (dr/dphi)^2. The calculation has the benefit of not leading to complex-valued results. The result is plotted in Fig. 9. The positive part now is on the lhs. The range is shifted to higher radii by the relativistic effects (H0 > H) but the result is less sensitive than in Fig. 8. Again we have only effects for the minimal radius of the ellipse, not the maximal radius.
Am 21.03.2017 um 11:49 schrieb EMyrone:
This is given by Eqs. (4) and (5), which can be solved by computer algebra to give the relativistic orbital differential function (dr/dphi) squared. In the non relativistic limit it reduces to Eq. (8). The all important advance made by computational solution of the relevant Euler Lagrange equations is that this differential orbital function is known now to be that of a precessing ellipse. So this ties up the solution and I will proceed to writing up Sections 1 and 2 of UFT373.