The A(t) Function or Correlation Function

These are generous remarks by GJE as ever. The A(t) function is a constant of integration in the sense that it does not depend on theta in a theta integration, must depend on time for there to be a three dimensional orbit. In order to model the evolution over time of a three dimensional orbit to a two dimensional orbit a correlation function is needed, indicating a background stochastic force in addition to the inverse square force. The Langevin equation and Debye theory give a plateau in the far infra red and fail completely. The memory function develops the friction coefficient and explains the far infra red. While working with the Brot group in CNRS Nice I learned of the rotational velocity correlation function which when Fourier transfromed gives the far infra red power absorption coefficent. This technique was applied in Omnia Opera 5 back at the EDCL while I was a Ph. D. student (Auto Two). That was the happiest time of my professional life apart from the last ten years of working with AIAS, the Ramsay Memorial Fellowship of 1976 to 1978 and a short interval at Cornell Theory Center (1988 – 1992). There is no doubt that both the spiral and the helix replicate themselves throughout nature, and we now know that the ellipse does the same, more accurately the precessing ellipse.

To: EMyrone@aol.com
Sent: 01/09/2014 22:25:59 GMT Daylight Time
Subj: Re: Beta Orbit with Finite A(t)

PS

What you may also be providing here is a fundamental new insight into the structure of dna (double helix held together by hydrogen bonds). In particular why there is a helical structure in the first place ( that would probably spiral outwards without the hydrogen bonds holding the two structures together).

Pauling got quite close to deciphering this structure as I recall from one of Mansel’s remarks ( before or about the same time as Watson and Crick). Point is, you may have “stumbled” on something very fundamental, all contained within the same theoretical framework, that is replicated in a number of ways throughout nature.

That a three dimensional orbit can become a planar two dimensional orbit over the course of time is an amazing discovery. Watson and Crick’s discovery deserved a Nobel Prize. So does yours ( and this is just one of many new discoveries)!!

Sent from Samsung Mobile

Subject: Beta Orbit with Finite A(t)

For an inverse square law of attraction this orbit is:

r = alpha / ( 1 + eps cos beta))

where:

beta = (L / L sub theta) theta + A(t)

The time dependent A(t) could be modelled with an exponential decay

A(t) = A(0) exp ( – t / tau)

where tau is a kind of relaxation time. if A(t) is assumed to be a correlation function:

<A(t)A(0)> / <A(0) squared> = exp (- t / tau)

then this would be Debye relaxation theory given by a Langevin equation, the stochastic force of which being the cosmic background force (Brownian motion force). The correlation function falls from unity to zero. When it reaches zero the orbit becomes planar and non precessing. Otherwise the orbit is always three dimensional. The Debye relaxation time tau must be obtained by measurement, it is of the order of millions to billions of years. There are many ideas like this that can be tried. The key point is that the orbit is a non precessing ellipse if an only if A(t) is zero, has relaxed away to zero over billions of years. At very short times the Debye theory fails completely (Omnia Opera first few papers) and is replaced by memory function theory. That could model what happens in the early stages of a galaxy or solar system orbit.

Discussion on Beta Orbit with Finite A(t)

Comment by GJE:

The A(t) function is indeed the key function for a three dimensional orbit, and this looks to be the most important discovery of UFT269 ff. Without it the the three dimensional orbit becomes two dimensional, even though spherical polar coordinates are being used, and even though L sub X, L sub Y and L sub Z are all non zero from fundamental geometry. So my idea here is to build on our Omnia Opera 20 on www.aias.us, the pioneering paper that led to the memory function technique, the first explanation of the far infra red absorption of liquids and similar materials, the D. Sc. degree, theHarrison Memorial Prize, and Meldola Medal of the Royal Society of Chemistry, all in two years (1978 and 1979). OO20 was written at Oxford and Aberystwyth. GJE took the far infra red data on a new interferometer granted to Rowlinson and myself at Oxford and set up at the EDCL, Room 262. I wrote the grant application with Prof. Sir John Rowlinson, F. R. S., my post doctoral supervisor at Oxford. This paper clinched GJE’s Ph. D. Thesis, more or less, and he became a University of Wales Fellow, Sloan Foundation Fellow and SERC Advanced Fellow in his own right. I was supervising his Ph. D. on behalf of Mansel Davies (Autobiography Volume Two above my coat of arms on the home page of www.aias.us). The original Faradey II paper in OO20 can be studied by just clicking on the Omnia Opera. Years of combined effort went in to constructing the Omnia Opera, which is archived every quarter at the British Library from the National Library of Wales.

To: EMyrone@aol.com
Sent: 01/09/2014 21:26:33 GMT Daylight Time
Subj: Re: Beta Orbit with Finite A(t)

Very interesting ideas and nature replicating itself.

Sent from Samsung Mobile

Beta Orbit with Finite A(t)

For an inverse square law of attraction this orbit is:

r = alpha / ( 1 + eps cos beta))

where:

beta = (L / L sub theta) theta + A(t)

The time dependent A(t) could be modelled with an exponential decay

A(t) = A(0) exp ( – t / tau)

where tau is a kind of relaxation time. if A(t) is assumed to be a correlation function:

<A(t)A(0)> / <A(0) squared> = exp (- t / tau)

then this would be Debye relaxation theory given by a Langevin equation, the stochastic force of which being the cosmic background force (Brownian motion force). The correlation function falls from unity to zero. When it reaches zero the orbit becomes planar and non precessing. Otherwise the orbit is always three dimensional. The Debye relaxation time tau must be obtained by measurement, it is of the order of millions to billions of years. There are many ideas like this that can be tried. The key point is that the orbit is a non precessing ellipse if an only if A(t) is zero, has relaxed away to zero over billions of years. At very short times the Debye theory fails completely (Omnia Opera first few papers) and is replaced by memory function theory. That could model what happens in the early stages of a galaxy or solar system orbit.

270(5): Checking Note 270(4)

The computer algebra by co author Horst Eckardt found a small alegbraic error in Note 270(4) which results in sin cubed theta being replaced by sin theta. This error does not affect the graphics posted yesterday, but to prepare for animation it results in Eq. (12) of this note, which should be checked once more by computer to make sure all is right before animation of the three dimensional orbit. The hyperbolic spiral is much easier to work with than the ellipse when it comes to animation because Eq. (12) gives phi as a function of t. For the ellipse it is possible analytically only to find t as a function of phi.

a270thpapernotes5.pdf

Daily Report Sunday 31/8/14

There were 1005 hits from 338 distinct visits, main spiders from google, MSN, yahoo, wotbox and yandex. 11 hours of recording on 1st September produced: Auto1 16, Auto2 2, F3(Sp) 8, UFT269 6, UFT88 6, Englynion 3, CEFE 3, Book of Scientometrics 3, Evans Equations 1 numerous Spanish, EGR Refutation Slides 1. AWStats top ten for Sept 1st: Auto1, UFT166, UFT269, CV, UFT88, UFT228, F3(Sp), Reduction to Maxwell, UFT37, Levitron. Excellent agreement between Webalizer and AWStats, indicating accurate stats. Landeszeitung Germany general; Denver University Autobiography Volume One; Massachusetts Institute of Technology (MIT) CV, D. Sc. Ceremony, Proofs One to Five, Summation Indices, general hypothesis for any metric, gravitational flow charts; Trade Union Confederation of Workers’ Commissions Spain F10(Sp); Academ Russia general; China extensive, United States National Archives general. Intense interst all sectors, updated usage file attached for 11 hours of 1st September 2014.

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Graphics for Note 270(4)

Excellent graphics, here we see the beginnings of a three dimensional galaxy. The three dimensional orbits transform to two dimensional orbits as A(t) goes to zero. So the function A(t) must be explored systematically and I will do this in future notes.

Three Dimensional Galactic Dynamics, Part 1

The surface r(theta,phi) gives a spiral on a cylindrical surface. The form spirals inward for phi –> infinity as usual for logarithmic spirals. The theta factor defines the hight of the cylindrical surface, see 3D plot and projection to XY axis.

Horst

EMyrone@aol.com hat am 1. September 2014 um 13:45 geschrieben:

This note defines the three dimensional hyperbolic spiral in Eq. (3), ans that can be graphed in a spherical polar plot. The 3 – D Binet equation shows that it is produced by an inverse cube law of attraction in three dimensions. Lagrangian dynamics gives its trajectory in Eq. (16), and this can be animated straightforwardly. In this case the inversion is not a problem. An infinite variety of three dimensional galaxies can be produced with this type of analysis, but it is known experimentally that the trajectory in a 2 – D whirlpool galaxy is the two dimensional hyperbolic spiral, so it is meaningful to extend that analysis to three dimensions.

Discussion of Note 270(4)

Eqs. (10) and (11) are obtained from Eqs. (6) and (7) using the lagrangian (1), in exactly the same way as for the ellipse, and same notation. So all OK.

In a message dated 01/09/2014 14:58:31 GMT Daylight Time, writes:

Did you obtain eq.(10) from differenciating beta given by (3)? Then a time derivative of theta seems to be missing, leading to

dbeta/dt = L/Lphi ( phi dot sin(theta) + phi theta dot cos(theta) )

How did you derive eq.(11)?

Horst

EMyrone@aol.com hat am 1. September 2014 um 13:45 geschrieben:

This note defines the three dimensional hyperbolic spiral in Eq. (3), ans that can be graphed in a spherical polar plot. The 3 – D Binet equation shows that it is produced by an inverse cube law of attraction in three dimensions. Lagrangian dynamics gives its trajectory in Eq. (16), and this can be animated straightforwardly. In this case the inversion is not a problem. An infinite variety of three dimensional galaxies can be produced with this type of analysis, but it is known experimentally that the trajectory in a 2 – D whirlpool galaxy is the two dimensional hyperbolic spiral, so it is meaningful to extend that analysis to three dimensions.

Literature Search

I did an extensive literature search on three dimensional orbits and there has been a great deal of work on them, not surprisingly, but none of that work has produced the incisive results of UFT269 and notes for UFT270. There has been a lot of work at NASA for example. The AIAS methodology is effective because ideas can be checked quickly, leading to a rapid development. Everything is discussed thoroughly and quickly. The mechanism of transition to planar orbits was not known prior to this work, and it was not known that 3 – D orbits produce precession, x = sin theta. There is a large real time following on the blog and the ECE sites. Obviously, standard physics missed almost all these important conclusions because it adheres to a completely failed theory of orbital precession, Einsteinian general relativity. It ignores scientific refutations of high quality, and ignores Baconian science. So its funding is being cut severely as is well known. What is the point of all that? None at all. The dogmatists just ignore the total failure of Einstein in whirlpool galaxies and go on claiming high precision. We now know that the high precision is due to Thomas precession, or if you wish, simple geometry.

Beta Orbit with Finite A(t)

For an inverse square law of attraction this orbit is:

r = alpha / ( 1 + eps cos beta))

where:

beta = (L / L sub theta) theta + A(t)

The time dependent A(t) could be modelled with an exponential decay

A(t) = A(0) exp ( – t / tau)

where tau is a kind of relaxation time. if A(t) is assumed to be a correlation function:

<A(t)A(0)> / <A(0) squared> = exp (- t / tau)

then this would be Debye relaxation theory given by a Langevin equation, the stochastic force of which being the cosmic background force (Brownian motion force). The correlation function falls from unity to zero. When it reaches zero the orbit becomes planar and non precessing. Otherwise the orbit is always three dimensional. The Debye relaxation time tau must be obtained by measurement, it is of the order of millions to billions of years. There are many ideas like this that can be tried. The key point is that the orbit is a non precessing ellipse if an only if A(t) is zero, has relaxed away to zero over billions of years. At very short times the Debye theory fails completely (Omnia Opera first few papers) and is replaced by memory function theory. That could model what happens in the early stages of a galaxy or solar system orbit.

Conservation of Angular Momentum in Three Dimensions

The three Cartesian components of angular momentum are given in 3 D by Eqs. (4) to (6) of Note 269(3). This hand calculation was checked by computer and is correct. Conservation of angular momentum means that all three components are conserved (dL sub X / dt = 0; dL sub Y / dt = 0; dL sub Z / dt =0). It is obvious that L sub X and L sub are not zero in general. The usual dogma assumes that they are zero. This result is true for any kind of radial force between m and M and is the result of pure geometry.

Discussion of Note 270(3)

I agree that it is seemingly planar in beta, but beta as a function of theta and phi. This gives Eq. (25), and the orbits that you graphed for UFT269. These are perfectly valid, showing precessing orbits in the function phi. This is the key difference from the two dimensional analysis in plane polar coordinates r and phi alone. That gives a non precessing orbit. The animation shows that the orbit is an intricate function of time, and is three dimensional. The three dimensional analysis reduces to the usual two dimensional analysis if A(t), the constant of integration in Eq. (23), is zero, or time independent. The Euler Lagrange equation (7) gives the same result as the fundamantal calculation of angular momentum from first principles in UFT269, so it valid. The Euler Lagrange equation (8) is also a standard equation. I checked this with a literature search. The Euler Lagrange equation (6) is valid because the lagrangian can be defined as in Eq. (1). The analysis is completely self consistent, cross checked with lagrangian and first principles. The integration constant A(t) in Eq. (23) must be time dependent, so it plays a role in evolution. It must also be theta independent. So A(t) can be regarded in analogy with relaxation theory and correlation function or diffusion theory. Initially it may be very large in the primaeval three dimensional orbit, but it falls to zero over billions of years, resulting in a planar orbit. The cosmic backgropund in your suggestion can be developed with relaxation theory, the background being the stochastic force that appears in Langevin or memory function theory.

In a message dated 01/09/2014 12:12:11 GMT Daylight Time,

My view is the following: Obviously it comes out from this note that an orbit with a central potential of type k/r is planar as also comes out from Euler-Lagrange theory. If such an orbit is considered in 3D, the plane of the orbit will be placed in an arbitrary angle compared to the XY plane. Nevertheless this remains a planar orbit. The beta angle obviously is the transformed phi angle of the XY plane which depends from phi and theta since the transformation needs two coordinates. So it would be illustrative to show that eq.(17) of the note is just this coordinate transformation, probably identical with the tetrad.

The Lagrange equations (7) and (8) seem to be unnecessary because Lagrangian theory requires coordinates without constraints as you know. The constraints have been applied to transform theta and phi to beta.

The time-dependent integration constant remains interesting. The meaning is in my opinion that an external acceleration is applied. This leads to a deviation from the k/r potential and, consequently, to non-planar orbits.

Horst

EMyrone@aol.com hat am 1. September 2014 um 11:25 geschrieben:

All planar orbits precess by observation, so they must be limits of three dimensional orbits. The Kepler Hooke Newton non precessing ellipse is obtained if and only if condition (17) of note 270(3) is true. The reason for a planar non precessing ellipse can be expressed mathematically , as in note 270(3), but there must still be a deeper reason due to physics, and that must be the result of evolution. To work out the evolution is a highly non trivial n mass problem needing supercomputers and additional assumptions. In the two mass problem the orbit is in general three dimensional, this is proven by the fact that all observable orbits precess. It is now known that the inverse square law produces this precession if the solution is correctly worked out with spherical rather than cylindrical polar coordinates. In Marion and Thornton for example, the entire orbital theory is worked out with cylindrical polar coordinates. The force law for the type of three dimensional orbits in galaxies is completely different from inverse square. In the roughly two dimensional whirlpool galaxies the orbit is a hyperbolic spiral which can be worked out with ECE theory, so one can start with that two dimensional spiral and make it three dimensional by changing phi to beta. That ought to produce some more very interesting graphics and animations. The Cartan torsion of the spherical polar coordinates can be worked out easily, and that alone is sufficient to show that this is a torsion based theory. So all orbits are in one sense due to torsion.