The work so far for UFT376 is very important and has made great progress. We need to make cross checks at this stage. With reference to Note 376(6) it is already known from your first class numerical work in UFT375 that orbital precession in the usual direction is given by the lagrangian (1). The Minkowski force equation for the orbit is obtained from this lagrangian, so the two methods must give the same result self consistently, the same precession and same direction of precession. This is a rigorous test of the code. Both the Minkowski force equation and the lagrangian are well known in special relativity as you know. The relativistic lagrangian is actually designed to give the relativistic momentum as in Marion and Thornton, using the canonical equation (2) of the note, which is the same as Marion and Thornton, third edition, chapter 14, Eq. (14.107). The relativistic momentum comes from conservation of momentum and the Lorentz transform. The Minkowski force is dp / d tau where tau is the proper time and where p is the relativistic momentum. So the fact that you have already obtained the orbital precession from the relativistic lagrangian in a previous paper (UFT375) means that exactly the same precession must emerge from the Minkowski force equation, the correctly relativistic Newton equation. The potential energy remains the same throughout. This assumes that the field Eq. (17) is decoupled from Eqs. (15) and (16). If Eqs. (15) to (17) are solved simultaneously (as they should be), and if kappa is used as an input parameter, one could possibly get precessions of all kinds, depending on a model for kappa. In addition, the precession from Eq. (1) must be the same as the result obtained in UFT328, using the scatter plot method and simultaneous solution of the same relativistic lagrangian and hamiltonian. This is another cross check. As you know, the difference between ECE2 and special relativity is that kappa appears in ECE2 together with the field equation (17). The kappa input parameter does not occur in special relativity. The precession cannot be different from the lagrangian and force equation, so the only possibility is that the code needs to be checked. Some Runge Kutta procedure may need a finer mesh or similar. Similarly, the non relativistic Newton equation is derivable from the non relativistic lagrangian, and both methods must give the same results.

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