425(2): New Equation for dm(r1) / dr1

425(2): New Equation for dm(r1) / dr1

The use of the Hamilton canonical equations of motion is shown in this note to lead to a new expression of dm(r1)/ dr1 in terms of the angular momentum L in Eq. (70).

dm(r1) / dr1 = – L squared / (c squared gamma squared m squared r1 cubed)

so m(r1) is constant unless there is an angular momentum present, another amazing result of m theory, although I say it myself. So the power of the Hamilton canonical equations becomes immediately apparent. It is clear that dm(r1) / dr1 is related to the centrifugal force L squared / (m r1 cubed). Hamilton derived his Hamiltonian dynamics in 1833 from the earlier Lagrangian dynamics, based in turn on work by Euler. Although he is always known as Rowan Hamilton in Trinity College Dublin, where I am sometime Visiting Academic, he became Sir William Rowan Hamilton, Irish Astronomer Royal, and a predecessor as Civil List Pensioner. He was offered F. R. S. but could not afford the fees. Clearly, he was not terribly interested in paying the fees, and the fact that he was not an F. R. S. does not mean a thing, the important things are the Hamilton dynamics, the hamiltonian, the Hamilton Principle of Least Action, his discovery of quaternions, and many other major discoveries. He was also a poet, not the greatest who ever lived, but nevertheless a poet. The entire subject of quantum mechanics is based on the hamiltonian, as is very well known.

a425thpapernotes2.pdf

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424(2): Geodesic Method of Deriving the Evans / Eckardt Equations

424(2): Geodesic Method of Deriving the Evans / Eckardt Equations

This was first and correctly derived in Eq. (23) of UFT416 and there was just a typo in the Note. So all is OK and I have completed the checks on the self consistency of the lagrangian and hamiltonian formulations of m theory on the classical level. After completing the present work with the Hamilton canonical equations I will go on to Schroedinger quantization of m theory. The Evans Eckardt equations allow a quantum force equation to be developed in m theory.

424(2): Geodesic Method of Deriving the Evans / Eckardt Equations
To: Myron Evans <myronevans123>

How did you derive eq.(12) for the linear momentum? From (11) I had expected an additional factor sqrt(m(r)) from r1.

Horst

Am 10.12.2018 um 10:32 schrieb Myron Evans:

424(2): Geodesic Method of Deriving the Evans / Eckardt Equations

This is the geodesic method first derived in UFT416. It shows in another way that the Evans Eckardt equations dH / dt = 0 and dL / dt = 0 are very fundamental and easier to use than the lagrangian method. As in 424(1) a lagrangian can be found to give the EE hamiltonian, and that leads to constraint equations with new information.

Note 424(1): Derivation of the Lagrangian of m Theory from the Hamiltonian

Note 424(1): Derivation of the Lagrangian of m Theory from the Hamiltonian

Thanks for going through this, the right method emerged in the latter part of Note 424(3), using the frame (r1, phi). This was used in the final Sections 1 and 2 of UFT424 posted on www.aias.us. The geodesic method of Note 424(2) was used to check the hamiltonian. Then in Note 425(1) a self consistent methodology was found, and now I am working on the Hamilton canonical equations for m theory. So it is now known that the lagrangian and hamiltonian of m theory are rigorously self consistent in frame (r1, phi), the frame of m space using the fundamental concepts of Euler Lagrange Hamilton dynamics. This is not true in the frame of flat space (r, phi).

Note 424(1): Derivation of the Lagrangian of m Theory from the Hamiltonian
To: Myron Evans <myronevans123>

The calculations are o.k. If see it right, the Lagrangian (19) should give the same equations as obtained from

dH/dt = 0,
dL/dt = 0.

Will check this. As I see this, the Hamiltonian is eq.(11) and the potential energy should contain the m(r) factor:

.
Angular momentum is

.

Horst

Am 08.12.2018 um 14:39 schrieb Myron Evans:

Note 424(1): Derivation of the Lagrangian of m Theory from the Hamiltonian

The lagrangian is Eq. (19), and is derived from the hamiltonian using the fundamental equation (1) of Lagrangian dynamics. The lagrangian is therefore rigorously equivalent to the hamiltonian and to the Evans Eckardt equations. This is an entirely new physics so a vast amount of new information is given by it. From now on I suggest using both hamiltonian and lagrangian methods. The first thing to do is to derive orbits, superluminal motion, energy from m space, and all previous results of UFT415 ff from the Evans Eckardt equations by solving dH / dt = 0 and dL / dt = directly without using the lagrangian. The rigorously correct lagrangian is now known and the Euler Lagrange equations (22) and (23) will give additional equations which must be equivalent to the Evans Eckart equations. This will lead to several new constraint equations such as (34). The rigorously correct lagrangian (19) reduces to the lagrangian used initially by inspection in the limit m ( r ) goes to one, but not identically equal to one. The Hamilton canonical equations can also be used at a later stage and quantization can be initiated. With the computer, any amount of complexity is no problem.

425(1): Self Consistent results from the Lagrangian and Hamiltonian Methods

425(1): Self Consistent results from the Lagrangian and Hamiltonian Methods

This note shows that the two methods are self consistent and produce a new equation (24) which is the generalization of the well known Eq. (25) of flat spacetime. For rigorous self consistency it follows that dm(r1) / dt = 0 and dm(r1) / dv1 = 0. This is because the fundamental infinitesimal line element and metric are those of a steady state universe. There is no expanding universe and no Big Bang. This was for example in UFT49. As shown UFT424, the fundamental equation (13) of Euler / Lagrange / Hamilton dynamics is true if and only of dm(r1) / dt = 0. All the results using the lagrangian theory of previous papers are rigorously correct: forward and retrograde precession, shrinking and expanding orbits, the possibility of superluminal motion, the possibilty of infinite energy from m theory, the description of the S1 star and the whirlpool galaxy, and in effect a completely new classical dynamics which overthrows the standard model on the classical level. Eq. (13) of fundamental Euler / Lagrange / Hamilton dynamics is obeyed rigorously by m theory, and the second Evans Eckardt equation dL / dt is given directly by both the classical method and the lagrangian method. The use of dH / dt = 0 and dL / dt = 0 gives Eq. (24), which gives new information on m(r1) and dm(r1) / dr1. The hamiltonian is given rigorously by the fundamental geodesic method, a Lagrangian method.

a425thpapernotes1.pdf

424(3) : Rigorous Self Consistency of the Hamiltonian and Lagrangian of m Theory

424(3) : Rigorous Self Consistency of the Hamiltonian and Lagrangian of m Theory

The frame is always that of the most general spherically symmetric spacetime. So one can just pick up any textbook and replace r by r1, essentially.

The Sagnac Effect in Frame (r1, phi)

The Sagnac Effect in Frame (r1, phi)

This is delta t = 4 pi r1 squared omega / c squared = 4 pi r squared omega / (m (r1) c squared) as in previous work Q. E. D. Experiments are needed to find m (r1). Any radial coordinate r in flat spacetime is replaced by r / m (r ) power half as in previous work. The Einstein theory produces m (r) = 1 – r0 / r and is totally wrong in the Sagnac effect. There are hundreds of new discoveries to be made by picking up a book on classical dynamics and rewriting it in m space.

424(3) : Rigorous Self Consistency of the Hamiltonian and Lagrangian of m Theory

424(3) : Rigorous Self Consistency of the Hamiltonian and Lagrangian of m Theory

This requires the use of the (r1, phi) frame, and the lagrangian of previous work, Eq. (48) or (49), is obtained from the hamiltonian of previous work, Eq. (34). All previous work, discoveries and concepts are rigorously correct: superluminal motion, infinite vacuum energy and the explanation of the S1 star, and much else. The (r1, phi) coordinate system must always be used in m theory, otherwise rigorous self consistency is lost. The (r1, phi) coordinate system also rigorously conserves the hamiltonian and angular momentum. This was shown using numerical methods by Horst Eckardt. We are now ready to move on to the quantum level. In this note I started out with the intent of analyzing constraint equations, but realized half way through that the use of the (r1, phi) system makes everything self consistent and there is no need for constraint equations. A completely new classical dynamics emerges. The lagrangian and hamiltonian are related by Eq. (48) so given the hamiltonian the lagrangian can be calculated, and vice versa. The Evans Eckardt equations must always be worked out in frame (r1, phi). In m space this is the fundamental frame.

a424thpapernotes3.pdf

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