Alpha Institute for Advanced Studies

427(1): Hamilton and Hamilton Jacobi equations for m Theory

Many thanks indeed Excellent computation and graphical presentations. This is another significant discovery in that the results are critically dependent on m(r1). So a very small departure from ECE2 (infinitesimal line element with m(r1) = 1) produces large effects within the Hamilton Jacobi formalism. The m theory is a new and complete system of dynamics, augmenting the Hamilton Jacobi system. As Horst mentions, this is reminiscent of chaos theory.

427(1): Hamilton and Hamilton Jacobi equations for m Theory

The calculations seem to be o.k. I computed the complete Hamilton equations for the inertial system (results o9-o12) and for polar coordinates (o14-o17) and additionally with abbreviation epsilon_1 (o19-o21). The m function appears in addition to the earlier relativistic results as expected.

I also solved the Hamilton-Jacobi equation (37). The resulting diff. eq. is o7 in section 5. This is not analytically solveable with a general m(r1). I graphed two solutions with m(r1)=1 and m(r1)=0.99, see attached figure. For m(r1)=1 the same result as found for UFT 426 comes out. I could simplify the solution method and avoid a quartic equation. For m(r1)=0.99 the results differ significantly, the orbital radius nearly doubles. We find as before that the results depend very critically on the form of m(r1).

Horst

Am 07.01.2019 um 11:09 schrieb Myron Evans:

427(1): Hamilton and Hamilton Jacobi equations for m Theory

This note shows that the Hamilton and Euler Lagrange equations produce precisely the same force from m space (spacetime, aether or vacuum), Eq. (22), which can become infinite under well define conditions as discussed in UFT417. The Hamilton Jacobi equations for m theory are equations (37) and (38), which can be integrated using Maxima to give the radial and angular actions. The quantum of action is h bar, the reduced Planck constant, so Schroedinger quantization of Eq. (37) leads to an entirely new relativistic quantum mechanics, which ought to be able to describe the Lamb shift in the H atom. The spin connection can be found from Eq. (22) as in previous work.

427(1).pdf

427(3): Quantization of m Theory in the H Atom

This is the first attempt at quantization of m theory following its rigorously consistent development on the classical level in immediately preceding notes and papers. The relativistic energy levels of the H atom in m theory are given by Eq. (16), in which the H atom wavefunctions can be used as a first approximation as in many previous papers. It is seen that the classical energy levels are all shifted to a different extent by m theory. It may also turn out from the computation that the energy levels are split, i.e that the well known multiple degeneracy of Schroedinger’s H is lifted. In other words the energy levels may become dependent on l and m as well as on n. Furthermore the m theory may eventually lead to a new explanation for the Lamb shift, which is a difference in energy between 2S sub half and 2P sub half. this is because the Lamb shift is caused by vacuum energy in the form of fluctuating electric and magnetic fields of the vacuum. In order to achieve that result however, the present first attempt must be refined to include spin the SU(2) basis. The m theory produces the spin connection and the vacuum force, so should be able to produce the Lamb shift. In fact the spin connection has already been used to describe the Lamb shift in recent UFT papers. So the concepts of ECE theory and m theory are rigorously self consistent over a span of 426 papers and books to date. About 300 have been translated into Spanish.

a427thpapernotes3.pdf

427(1): Hamilton and Hamilton Jacobi equations for m Theory

This note shows that the Hamilton and Euler Lagrange equations produce precisely the same force from m space (spacetime, aether or vacuum), Eq. (22), which can become infinite under well define conditions as discussed in UFT417. The Hamilton Jacobi equations for m theory are equations (37) and (38), which can be integrated using Maxima to give the radial and angular actions. The quantum of action is h bar, the reduced Planck constant, so Schroedinger quantization of Eq. (37) leads to an entirely new relativistic quantum mechanics, which ought to be able to describe the Lamb shift in the H atom. The spin connection can be found from Eq. (22) as in previous work.

a427thpapernotes1.pdf

The mathematics of Cartan geometry used in ECE is exactly the same as used by everyone else, for example Sean Carroll, a prominent theoretical physicist, who has described ECE as a valid theory. This alone is enough to show that wikipedia is defamatory. A fraudster like Bruhn does not attack Carroll, who uses the same mathematics exactly as I do. On close examination we found that Bruhn is a fraudster, in that he attacks well known mathematics using what appear to be deliberate distortions. So his site is nonsense mathematics. Wikipedia is in league with Bruhn, this is well known. Wikipedia has no credibility when it comes to ECE theory and its referrals have been stuck on one figure for nearly a decade. This means that interest in the Wikipedia defamation evaporated a decade ago. People are not stupid and know a witchhunt when they see one. McCarthy became the most hated Senator in history and was finally censured by the Senate as is well known. The www.aias.us site gets millions of referrals. The crackpot site has 32 referrals out of several million, and that figure has been stuck for several years while the referrals move on every day as recorded in my early morning daily report. Wikipedia also distorts the Nobel nominated B(3) field out of all recognition by omitting all mention of the fact that I have been nominated a few times for a Nobel Prize for the inference of B(3). This gross defamation has become well known, so wikipedia has imploded, its credibility is zero. Nonetheless it would be optimal if all the dirt could be removed from the internet. Crackpot is loathed like Jo McCarthy as we know.

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

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.

a426thpapernotes6.pdf

Evaluation of Hamilton equations by computer

This is indeed an important result, and congratulations! Effectively it gives a rigorous theoretical explanation for light deflection by gravitation using the Hamilton equations in special relativity, and a theoretical justification for the explanation of light deflection by gravitation in previous UFT papers. As you know, it was shown in UFT150 – UFT155 that the Einsteinian explanation is riddled with errors and obscurities, and is effectively pseudoscience. UFT150 – UFT155 are now classics, and cannot be covered up. This work of yours is spectacular proof that the Hamilton equations give important information in special relativity. Solving for gamma is a key idea, so it becomes possible to choose p = gamma m v sub N as a generalized coordinate, and q = r. You have also shown that the first order Hamilton equations can be integrated directly by the Maxima routine, another major advantage. The Hamilton equations of 183 were first discovered by Lagrange in 1809, but attributed to Hamilton because he derived them using his Principle of Least Action. Evaluation of Hamilton equations by computer

The Hamilton equations can be evaluted by computer. The equations can directly be given as input since they are of first order as the Runge-Kutta solver requires.
I have investigated the definition of the generalized gamma which – formulated with rel. p – is a function gamma=f(gamma,v). It is possible to resolve this equation for gamma, giving two solutions which are graphed in the protocol gamma.pdf. The second solution seems to be unphysical, the first has a pole at v=c/2. This gave me the idea to compare it to the gamma of photons which we derived as

gamma_ph =

.

The results are similar with a streching of the v axis. Therefore I introduced a factor in the definition of gamma as worked out in the second protocol gamma-4.pdf:

.
This gives two solutions again, and the second is identical to that for photons! We have gamma(v=c) = sqrt(2). I find this a remarkable result, bringing together different paths of our development.

Horst

Am 03.01.2019 um 10:28 schrieb Myron Evans:

Evaluation of Hamilton equations by computer

This is an excellent computer analysis of the Hamilton equations. In order to test the various results they can be compared with the orbits from the Lagrange equations and EE equations in the previous UFT papers. The Hamilton equations must give the same orbits as the Lagrange equations and Evans Eckardt equations. The EE equations are the most fundamental and powerful to date because of the ability of Maxima to integrate them. Can the Hamilton equations be integrated numerically? The lagrangian (L) can be computed from the fundamental L = p q dot – H for each choice of p and q in the protocol. This should lead to the lagrangian for special relativity used in previous papers. These results can also be compared with the Evans Eckardt equations for special relativity: dH / dt = 0 and dL / dt = 0, where H = gamma m c squared – mMG / r and L = gamma m r squared phi dot. In order to reduce correctly to classical theory only one choice of lagrangian and only one choice of hamiltonian is possible.

gamma.pdf
gamma-4.pdf

Corrigendum Re: [] Fwd: Evaluation of Hamilton equations by computer

OK thanks, the overall result is unaffected, p and q can be defined as gamma m v sub N and r in the Hamilton equations, where there is a freedom of choice of p and q. We know from the Lagrangian analysis that p = gamma m v sub N, the relativistic momentum of special relativity. Physics is changing very rapidly.

Corrigendum Re: [] Fwd: Evaluation of Hamilton equations by computer

Here comes the corrected graphics. The functions are not identical. There is essentially an additional factor of c^2/v^2 in the self-consistently determined gamma.

Horst

Am 03.01.2019 um 15:21 schrieb Horst Eckardt:

I had an error in the third graph of gamma-4.pdf. The curves for photons is not identical to that of Hamilton theory, but the asymptotic values are identical.

Horst

Am 03.01.2019 um 15:01 schrieb Horst Eckardt:

The Hamilton equations can be evaluted by computer. The equations can directly be given as input since they are of first order as the Runge-Kutta solver requires.
I have investigated the definition of the generalized gamma which – formulated with rel. p – is a function gamma=f(gamma,v). It is possible to resolve this equation for gamma, giving two solutions which are graphed in the protocol gamma.pdf. The second solution seems to be unphysical, the first has a pole at v=c/2. This gave me the idea to compare it to the gamma of photons which we derived as

gamma_ph =

.

The results are similar with a streching of the v axis. Therefore I introduced a factor in the definition of gamma as worked out in the second protocol gamma-4.pdf:

.
This gives two solutions again, and the second is identical to that for photons! We have gamma(v=c) = sqrt(2). I find this a remarkable result, bringing together different paths of our development.

Horst

Am 03.01.2019 um 10:28 schrieb Myron Evans:

Evaluation of Hamilton equations by computer

This is an excellent computer analysis of the Hamilton equations. In order to test the various results they can be compared with the orbits from the Lagrange equations and EE equations in the previous UFT papers. The Hamilton equations must give the same orbits as the Lagrange equations and Evans Eckardt equations. The EE equations are the most fundamental and powerful to date because of the ability of Maxima to integrate them. Can the Hamilton equations be integrated numerically? The lagrangian (L) can be computed from the fundamental L = p q dot – H for each choice of p and q in the protocol. This should lead to the lagrangian for special relativity used in previous papers. These results can also be compared with the Evans Eckardt equations for special relativity: dH / dt = 0 and dL / dt = 0, where H = gamma m c squared – mMG / r and L = gamma m r squared phi dot. In order to reduce correctly to classical theory only one choice of lagrangian and only one choice of hamiltonian is possible.

gamma-4.pdf

New Formulation of the Hamilton Equations in Special Relativity

This uses the choice (20) of canonically conjugate generalized coordinates and the hamiltonian (33). The choice (20) means that there appear two new equations of motion, Eqs. (22) and (23), which may be solved numerically with the Evans Eckardt equations (25) and (26). It is shown that the Hamilton equations produce the force equation (37) of special relativity, used in previous UFT papers to produce a precessing orbit. So both the Hamilton and Euler Lagrange equations produce the same force equation. In addition, the Hamilton equations give the new equations of motion (22) and (23). To deduce these analytically requires the complete consideration of Euler Lagrange Hamilton dynamics. So the Hamilton equations produce new information not given by the Euler Lagrange equations. Finally the choice of canonically conjugate generalized coordinates (4). when used in the Hamilton equation (41) shows that the angular momentum is a constant of motion. In the Hamilton Jacobi formalism it is therefore defined by the action equations (46) and (47).The Hamilton Jacobi formalism therefore defines the action. The three complete systems of dynamics were traditionally the Euler Lagrange, Hamilton and Hamilton, but a fourth system has just been inferred, the Evans Eckardt system of dynamics. All four should be used together for any problem being considered. Their power is greatly amplified by the ability of the Maxima code written by Horst Eckardt to integrate them.

a426thpapernotes5.pdf

Evaluation of Hamilton equations by computer

This is an excellent computer analysis of the Hamilton equations. In order to test the various results they can be compared with the orbits from the Lagrange equations and EE equations in the previous UFT papers. The Hamilton equations must give the same orbits as the Lagrange equations and Evans Eckardt equations. The EE equations are the most fundamental and powerful to date because of the ability of Maxima to integrate them. Can the Hamilton equations be integrated numerically? The lagrangian (L) can be computed from the fundamental L = p q dot – H for each choice of p and q in the protocol. This should lead to the lagrangian for special relativity used in previous papers. These results can also be compared with the Evans Eckardt equations for special relativity: dH / dt = 0 and dL / dt = 0, where H = gamma m c squared – mMG / r and L = gamma m r squared phi dot. In order to reduce correctly to classical theory only one choice of lagrangian and only one choice of hamiltonian is possible.

Hamilton-2.pdf