Many thanks, this is good progress towards an analytical solution. I suggest comparing Eq. (16) with Eq. (20) using Eq. (27), and using H0 and L as constants of motion in the input parameters of the code. H0 can be replaced by – mMG /2a) where a is the semi major axis, a = alpha / (1 – eps squared), and L squared = alpha m squared MG. The astronomical observables are alpha and epsilon as you know. We are down to basics now because we are simply comparing
H (non rel) = 1/2 m v squared + U
H (relativistic) = (gamma – 1) mc squared + U
where U = – mMG / r. We have
gamma = ( 1 – (v/c) squared) power – half
with v squared given by Eq. (14) of the note. Finally compare dr / dphi in both cases (relativistic and non relativistic). The difference can be plotted, and a method can be found of finding the same quantity from astronomical observation. From your excellent numerical work with the lagrangian method, it is known that the difference represents a precessing orbit, a major discovery in my opinion, one that makes much of standard gravitational physics entirely obsolete, even more entirely obsolete than before. In general I am interested in applying the lagrangian method to a series of well known precessions, without using Einsteinian general relativity of course, de Sitter, geodetic, Thomas precession and so on. This analytical work simply rounds off the numerical discovery that orbital precession can be inferred using just the lagrangian of ECE2 relativity, which is special relativity in a space of finite torsion and curvature. The analytical work is the icing on the cake. The big breakthrough is that the Maxima method can solve very complicated simultaneous Euler Lagrange equations, so all the most recent papers are very popular.
Sent: 20/03/2017 21:42:23 GMT Standard Time
Subj: Re: Note 373(5): Direct Integration of the Hamiltonian
Fig. 3 shows the r(phi), dr/dphi and (dr/dphi)^2 for a Newtonian orbit. With relativistic correction the zero crossings of dr/dphi should be shifted to the right.
I compared eq.(19) with eq.(26). I plotted both terms for (dr/dphi)^2 and their difference in Fig. 4. Both terms should be positive because they are squared but the relativistic term (26) is not. There is a region of imaginary values dr/dphi. The bigger effect should be at phi=pi, but it is at phi=2 pi. This seems not to be very satisfactory.
Am 20.03.2017 um 15:28 schrieb EMyrone:
The direct integration produces the orbital differential function (16), which is known from the numerical lagrangian analysis to be that of a precessing ellipse. This function (16) can be compared directly with the non relativistic result expressed in Eqs. (20) and (27). The deviation of the relativistic orbit from the non relativistic orbit characterizes a precessing orbit. This is known from the fact that a numerical lagrangian analysis of ECE2 relativity gives a precessing orbit. There are no additional assumptions at all used in this simple analytical theory. So it is possible to deduce how a precessing orbit can be described by the differential orbital function dr / dphi. This by passes the need for numerical integration. So the exact precessing orbit is given by dr / dphi, an important result that is easy to compute and graph. The same function dphi / dr can be obtained numerically from the lagrangian. It should be possible now to make a direct comparison of this theory with the astronomical data on precession.