This is an elegant result, the non relativistic limit of which is the Newtonian r double dot = – MG r / r cubed. The computer algebra has revealed a structure that was not obviously present in the initial lagrangian. The structure of this equation can be compared with the structure of the equations in UFT374, which also give orbital precession. It would be interesting to graph the precessing three dimensional orbits. This theory can now be applied to all known relativistic precessions, for example: geodetic or de Sitter precession, Thomas precession, Lense Thirring precession, and equinoctial precession. A shrinking orbit may be produced by an appropriate potential energy, for example a non central potential energy. This type of theory should also produce radiational deflection due to gravitation. Finally, this result is expressible in any coordinate system.

To: EMyrone@aol.com

Sent: 10/04/2017 11:43:02 GMT Daylight Time

Subj: Re: Computation of Precessing Ellipse in Cartesian CoordinatesThe Euler-Lagrange equations take a clearly represented form in Cartesian coordinates:

(Index d stands for dot again). This scheme can easily be extended for 3 dimensions and written in vector form. The gamma factor also pertains to the potential energy term. The additional term at the rhs is of order 1/c^2 and is

bold v (bold r * bold v) / (c^2 r^3).

Horst

Am 10.04.2017 um 10:21 schrieb EMyrone:

Many thanks! This is another very important result. It is possible to obtain orbital precession in Cartesian coordinates with the ECE2 lagrangian, making Einsteinian general relativity redundant. It is also very interesting to see that the angular momentum is not a constant of motion in Cartesian coordinates. It is a constant of motion only in plane polar coordinates. The relativistic angular momentum is

L = gamma L0

where gamma is the Lorentz factor and L0 the classical non relativistic angular momentum. This can be coded in to find its effect on the orbit. I can google around to try to find the optimal experimental data for comparison. I found one such system where m orbits M with a precessions about thirty degrees per orbit, compared with fractions of seconds of arc in the solar system.

To: EMyrone

Sent: 09/04/2017 16:00:14 GMT Daylight Time

Subj: Re: 375(3): Relativistic Lagrangian in Cartesain CoordinatesFor strong relativistic effects, we obtain a precessing ellipse also in Cartesian coordinates as expected, see Fig. 1. Please notice that eq.(6) of the note only holds in the non-relativistic case. This term for L is not constant, see Fig. 2.

Horst

Am 09.04.2017 um 15:53 schrieb EMyrone:

This is given by Eq. (1), and simultaneous numerical solution of Eqs. (3), (4) and (6) should be a precessing ellipse.