Alpha Institute for Advanced Studies

425(3): Rigorous Self Consistency of m Theory

This note is a detailed demonstration of the rigorous self consistency of m theory in the elegant Euler Lagrange Hamilton dynamics. This is the first time that detailed consideration has been made of Hamilton’s equations in the UFT series. The self consistent choice of the Hamilton canonical variables is given in Eqs. (27) and (28). This gives the Einstein energy equation (36) in m space, first derived in UFT424. Eq. (47) is obtained in a rigorously self consistent manner from both Eqs. (42) and (43). This checks the starting equation of Note 425(2). The static solution is rigorously equivalent to the rest energy (56) of m theory. The static solution is for a particle m at rest attracted gravitationally by a particle M at rest in m space. The general solution is Eq. (57) and in the inertial frame gives the remarkable result (68) for the hamiltonian, reducing it to classical format in m space. The first Evans Eckart equation reduces this to the force equation in m space in an inertial frame, Eq. (76), giving a new definition of vacuum force, Eq. (80). The hamiltonian can be transformed to plane polar coordinates (r1, phi) giving Eq. (88). Finally the solution obtained in Note 425(2), and checked by computer algebra, is given in Eq. (93) in plane polar coordinates (r1, phi). This related dm(r1) / dr1 to M(r1). This is very complicated equation but can be solved as discussed by Horst this morning. The final note for UFT425 will be considered in the next and final note, using the second Hamilton equation. This is an entirely new classical dynamics valid for any m space.

a425thpapernotes3.pdf

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

Many thanks for this interesting protocol. This is an interesting and important problem and to eliminate any possibility of human error the equations can be solved entirely by computer. The fundamental equation being used is partial L / partial r = – partial H /partial r. The calculation depends on the choice of conjugate generalized coordinates p and q in Hamiltonian dynamics. If relativistic momentum p and r have are chosen as canonical variables, partial p / partial r = 0. In Lagrangian dynamics the generalized coordinates q dot and q are conjugate, so in Lagrangian dynamics partial q dot / partial q = 0. We have H = p q dot – L where H is the hamiltonian and L the lagrangian (Eq. (6.153) of Marion and Thornton, third edition). From the Hamilton and Lagrange equations p dot = – partial H / partial q = partial L / partial q. So the above equation is obtained with q = r. It follows that partial (p q dot) / partial q = 0. These are the fundamental equations being used, in the frame (r1, phi), with: H = m(r1) gamma m c squared – mMg / r1
L = – m c squared / gamma + mMG / r1
gamma = (m(r1) – (r1 dot squared + r1 squared phi dot squared)) / c squared) power minus half

I have made a few more calculations by hand and will send them around today, making a precise definition of the problem for the computer to take over to calculate everything from beginning to end, coming up with various important solutions. In Eq. (67) of the last Note, one possibility for constant m(r1) is dphi / dt = 0 for any m(r1). That gets rid of the problem of imaginary solutions that you mention below.

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

It is a good hint that eq.(66) could be used to determine m(r1). This diff. eq. cannot be solved by Maxima, only partially giving a log expression for m, see eq. E57e. This could be used as a starting point for an iterative solution but one has to keep in mind that the trajectories r1 and phi have to be computed a priori from a given m(r1). This would define a self-consistency cycle as is used for atomic physics where the exact potential is not known a priori.
I did some tests with m=const, however I arrived at imaginary gamma’s for m=1, here m=-1 seems to be required. Maybe something is still wrong in the calculations.

Horst

Am 19.12.2018 um 07:58 schrieb Myron Evans:

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

Many thanks again, the computer check is a great help as usual. I went through the protocol and its O14 is the same as Eq. (65), so a factor 2 must be put in front of the right hand side of Eq. (66). This was just a typo. Eq. (67) is another human error, the right hand side of Eq. (67) should contain the factor (1 + gamma squared m (r1)) / (gamma squared m(r1) – 1), and this should appear in Eq. (70). In the limit of flat spacetime, dm(r1) / dr1 is zero in Eq. (66), so in this limit dphi / dt = 0, and v sub N = r1 dot, the inertial frame. So a lot of new information, and equations, emerge from the use of the Hamilton equations. Many more developments are now possible. Eq. (66) is a differential equation for m(r1), so m(r1) it is no longer empirical. The other Hamilton equation gives a new equation, and there are also vector Hamilton equations. I am working on this now. Finally I checked that the Hamilton equations work fine for special relativity, or flat space. They are of course related to the Evans Eckardt equations and Euler Lagrange equations. The great elegance of classical dynamics can only be appreciated if all the available equations are used.
New Equation for dm(r1) / dr1

I could reproduce eq.(66), with a missing factor of 2. How did you arrive at 67? Where did the m(r1) functions go?

Horst

Am 16.12.2018 um 13:56 schrieb Myron Evans:

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.

425(2)-a.pdf

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

Many thanks again, the computer check is a great help as usual. I went through the protocol and its O14 is the same as Eq. (65), so a factor 2 must be put in front of the right hand side of Eq. (66). This was just a typo. Eq. (67) is another human error, the right hand side of Eq. (67) should contain the factor (1 + gamma squared m (r1)) / (gamma squared m(r1) – 1), and this should appear in Eq. (70). In the limit of flat spacetime, dm(r1) / dr1 is zero in Eq. (66), so in this limit dphi / dt = 0, and v sub N = r1 dot, the inertial frame. So a lot of new information, and equations, emerge from the use of the Hamilton equations. Many more developments are now possible. Eq. (66) is a differential equation for m(r1), so m(r1) it is no longer empirical. The other Hamilton equation gives a new equation, and there are also vector Hamilton equations. I am working on this now. Finally I checked that the Hamilton equations work fine for special relativity, or flat space. They are of course related to the Evans Eckardt equations and Euler Lagrange equations. The great elegance of classical dynamics can only be appreciated if all the available equations are used.
New Equation for dm(r1) / dr1

I could reproduce eq.(66), with a missing factor of 2. How did you arrive at 67? Where did the m(r1) functions go?

Horst

Am 16.12.2018 um 13:56 schrieb Myron Evans:

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.

425(2).pdf

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

OK many thanks! This is true, but there is another solution: m(r1) gamma = 1 / gamma, so m(r1) = 1 / gamma squared. This is the v sub N = 0 solution for a particle at rest. In general the Hamilton and Euler Lagrange equations hold for any coordinate system, so I proceeded with the plane polar system in Eqs. (58) and (59) as you note. This leads to a solution for an orbit. Eq. (57) is the fundamental Eq. (55) of Euler Lagrange Hamilton dynamics, so is always valid in any coordinate system.
425(2): New Equation for dm(r1) / dr1

I encountered a problem here: Eqs. (56,57) hold in the inertial coordinate system where gamma does not depend on r1. Therefore the rhs of both equations should be zero. Furthermore, it follows from the lhs that

partial m(r1) / partial r1 = 0.

The situation is different in the (r1, phi) system. Eqs. (60/61) are correct but the question is if you can use (57) in this case. Maybe you can. Then the subsequent calculation is valid. I will try to obtain the result (70) by computer.

Horst

Am 16.12.2018 um 13:56 schrieb Myron Evans:

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.

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

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

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

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

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)

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.