426(6): Hamilton Jacobi Equation for Special Relativity and Rigorous Self Consistency

Agreed with the first point, and the computation of the action is indeed a wholly original result, and congratulations on the computation! I do not think that the Hamilton equations have been applied to special relativity prior to this work, neither have the Hamilton Jacobi Equations, which have ramifications for quantization and for other areas of physics. The results for the action are wholly original, and an important step towards the application of the Hamilton Jacobi equation in m theory. These discussions of the renowned Evans Eckardt dialogue (circa 2005 to present) show clearly how the concepts, ideas and algebra are constantly checked and rechecked hundreds of times.
426(6): Hamilton Jacobi Equation for Special Relativity and Rigorous Self Consistency

The transition from the first line to the second line of eq.(5) is only understandable with the protocol I sent over today morning. We know that the transition is correct.
I succeeded in computing the action function S_r for the central motion problem (19). First the equation had to be resolved for (partial S_r/partial r)^2. This then gives a quartic equation fo partial S_r/partial r. Solving this gives four diff. equations which look similar:

The solutions are analytical and higly depend on relations between the parameters. I therefore put in the parameters of the numerical model calculations. The total energy including the term mc^2 has to be used. One obtains four partially complex functions. The real parts have been plotted in the Figure. One sees that these are (besides a null function) exactly two inverse functions, probably describing the two possible directions of motion of the orbiting mass. The functions are non-constant exactly in the physical range of r which in this example is 0.3 < r < 1. This is probably the first time the action function S_r was determind for the central motion problem.

Horst

However one has to insert

Am 05.01.2019 um 10:00 schrieb Myron Evans:

426(6): Hamilton Jacobi Equation for Special Relativity and Rigorous Self Consistency

This note defines the hamiltonian as Eq. (1), as in the Sommerfeld equation, and shows that the Hamilton and Lagrange methods give the same equations of motion for special relativity. These are written in the inertial frame and in plane polar coordinates. So the orbital precession becomes the same in both formalisms using this new method. From the special relativistic hamiltonian (14) the Hamilton Jacobi equations (19) and (20) are derived. They can be solved to give the actions for translational and rotational motions by integrating Eqs. (19) and (20). The quantized action is the angular momentum h bar, so the Hamilton Jacobi equation is a route to quantization. The next and final step is to develop the Hamilton Jacobi equations for m theory.

426(6).pdf


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