I reviewed these basics as described by Marion and Thornton. The number of proper Lagrange variables is equal to the number of degrees of freedom, but it is also possible to use the method of undetermined multipliers when the number of Lagrange variables is greater than the number of degrees of freedom. However, a simple solution to the problem is to solve Eqs. (17) to (19) of Note 371(2) simultaneously, to give r1, r2 and r3 in terms of theta, phi and chi. These are the required orbits. There are three dimensions (degree of freedom) and three proper Lagrange variables, r1, r2, and r3. So there are three differential equations in three unknowns, an exactly determined problem. The orbits are r1(theta, phi, chi), r2(theta, phi, chi) and r3(theta, phi, chi). Finally use

r squared = r1 squared + r2 squared + r3 squared

to find r(theta, phi, chi) and its precessions. In the planar limit it should reduce to a conic section without precession.

This is indeed an important paper, “importance” can be measured with objectivity with the scientometrics. Popularity is not always a good indicator, but more often than not it is. readership is orders of magnitude more immediate and meaninglul than citations. The dialogues and discussions that accompany each note are of equal importance, because the clarify basic concepts. Rotational dynamics is a complicated subject and discussions and graphics help clarify the concepts. The gyroscope dynamics are exceedingly intricate and are clarified by the computer. To resolve the latest discussion I am very curious to see what happens when Eqs. (17) to (22) are solved simultaneously. If the results are reasonable then the concepts are reasonable. If the results are meaningless or ill behaved in some way then the concepts need to be simplified. this is all part of the methodology being followed worldwide by a huge readership.

To: EMyrone@aol.com
Sent: 25/02/2017 09:26:44 GMT Standard Time
Subj: Re: FOR POSTING: Section 3 of paper 369

Many thanks for your eulogising comments. The method of numerical solution of the complete set of gyro equations is indeed very powerfull and applicable for cases I never thougth of.

Horst

Am 25.02.2017 um 10:18 schrieb EMyrone:

These results are full of interest and will attract a large readership. I can see that through the scientometrics. These graphics and analyses are clearly thought out as usual, and the results are interpreted so that the readership will be able to understand them without ploughing through the maths if they want to concentrate on engineering essentials. The results also show the great power of the Maxima code, controlled by the skill and experience of co author Dr Horst Eckardt. A huge amount of new information comes out of a problem that up to a few week ago was only vaguely understood, half understood and misinterpreted. Having understood the problem at last, all kinds of engineering solutions become possible, so we can engineer new kinds of railway systems for example with reduced drag. In the aerospace industry these results can eb used to enginner new kinds of mechanism that will allow aircraft to lift off, helping the wings lift the plane. These mechanisms can also be used in vehicles of all kinds to reduce drag. This is an outstanding section. It shows what can be done with the elegant mathematics of the eighteenth century enlightenment. A large number of possibilities has suddenly emerged.

Sent: 24/02/2017 19:02:41 GMT Standard Time
Subj: Section 3 of paper 369

I finished section 3 of paper 369. I gave several examples for gyro
motion, including application of an external torque. I could not verify
the Shipov experiment (lifting a spinning top by a torque in Z
direction). This seems to be an effect of different origin, maybe a
fluid dynamics effect of spacetime, although it is astonishing that it
is of macroscopic order.

Horst

The point mass M is placed at the origin and the point mass m is at (r1, r2, r3). The motion of the axes e1, e2, and e3 is defined with the dynamics of the Euler angles through the spin connection and equations (6) to (8). This gives six equations in six unknowns, Eqs. (17) to (22). That is an exactly determined problem therefore. The three degrees of freedom are the three dimensions of three dimensional space. Only three coordinates are being used, e sub 1, e sub 2 and e sub 3, and six Lagrange variables, r1, r2, r3, theta, phi and chi. The Euler equations for a rigid object are not being used, because there are no moments of inertia being used. Have you tried running these six simultaneous equations through Maxima, to find how the system behaves? If it gives reasonable results all looks OK. In spherical polar coordinates the mass M is at the origin and the mass m is at vector r. Solutions in the spherical polar system are also full of interest and much less complicated. It is also possible to use your idea of a unit vector by adapting the gyroscope with one point fixed. There are many interesting things to work on adn they will all create a lot of interest. It is best to work with spherical polar coordinates by Ockham’s Razor, but the Euler angles give a very large amount of new information. The important thing is the ability of Maxima so solve very complicated sets of simultaneous differential equations.

To: EMyrone@aol.com
Sent: 25/02/2017 09:09:39 GMT Standard Time
Subj: Re: Discussion of Note 371(3) and 371(2)

Your argument on time-dependence of the (e1, e2, e3) frame is correct. However it is not possble to describe a probelm with 3 degreees of freedom by 6 coordinates, at least not in Lagrange theory. Then you obtain an underdetermined system of equations, there is no unique solution.
A second point is to strictly discern if a masspoint or a rigid body is considered. You cannot use the Lagrange equations for a rigid body and apply it to a masspoint because then you have too many coordinates.
On the other hand it is possible to use Eulerian angles for masspoints. However you have already 3 coordinates so you cannot introduce additional translations. This type of application seems to be restricted to pure rotations on a unit sphere. The situation is different for a rigid body again.

Horst

Am 25.02.2017 um 09:53 schrieb EMyrone:

It is a good idea to use the gyro with one point fixed for orbital theory, the mass M is at the fixed point of the gyro, and mass m is separated by a distance r from M. Unlike problem 10.10 of Marion and Thornton, however, the distance r is not constant. I will look in to this and go back to the basics of the derivation of the Euler equation from variational calculus, Marion and Thornton chapter five (Euler 1744). Howevber, I think that all is OK for the following reasons. It is true that the Euler angles relating frame (i, j, k) and (e1, e2, e3) are constants by definition, provided that frame (e1, e2, e3) always has the same orientation with respect to frame (i, j, k) and provided that the two frames are static. Then theta, phi and chi, being constants, cannot be used as variables. I agree about this point. However in Note 371(2), frame (e1, e2, e3) is moving with respect to (i. j. k), and so theta, phi and chi are also moving. This is because the Cartesain frame (i, jk, k) is static by definition, but frame (e1, e2, e3) is dynamic, i.e. e1, e2 and e3 depend on time, but i, j, and k do not depend on time. Similarly in spherical polars, (i, j, k) static, but (e sub r, e sub theta, e sub phi) is time dependent so r, theta and phi all depend on time. The lagrangian (1) of Note 371(2) is true for any definition of v, and Eq. (10) of that note is true for any definition of the spin connection (e.g. plane polar, spherical polar, Eulerian, and any curvilinear coordinate system in three dimensions). So Eqs. (11) to (16) of that note are correct. So it is correct to set up the lagrangian (16) using the Lagrange variables r1(t), r2(t), r3(t), theta(t), phi(t) and chi(t). It could also be set up with plane polar or spherical polar coordinates. We have already correctly solved those problems using the lagrangian method, The fundamental property of the spin connection is to show how the axes themselves move and it is valid to re express the angles of the plane polar and spherical polar coordinates as Eulerian angles. In the orbital problem, the Eulerian angles are all time dependent. So they vary in this sense, and can be used as Lagrange variables. The Euler variable x of chapter five of Marion and Thornton is t, x = t. To sum up, it is true that the Euler angles are constants when viewed as angles defining the orientation of a static (1, 2, 3) with respect to a static (X, Y, Z), but in Eq. (10) of Note 371(2), the components of the spin conenction are defined in terms of time dependent Euler angles, which are therefore Lagrange variables. This fact can be seen from Eqs. (6) to (8) of the note, in which appear phi dot, theta dot, and chi dot. These are in general non zero, i.e. they are all time dependent angular velocities.

To: EMyrone
Sent: 24/02/2017 18:35:19 GMT Standard Time
Subj: Re: Note 371(3) : Definition of Reference Frames

Thanks, this clarifies the subject. I think we must be careful in applying Lagrange theory. A mass point in 3D is described by 3 variables that are either [X. Y. Z] or [r1, r2, r3]. The Eulerian angles describe the coordinate transformation between both frames of reference bold [i, j, k] and bold [e1, e2, e3]. The Eulerian angles are the sam for all points [r1, r2, r3]. So they cannot be subject to variation in the Lagrange mechanism. The degree of freedom must be the same in both frames, otherwise we are not dealing with generalized coordinates and Lagrange theory cannot be applied.

A Lagrange approach in Eulerian coordinates can be made if we describe the motion of a mass point that is described as a unit vector in the [e1, e2, e3] frame. Then [r1, r2, r3] is fixed and the Eulerian angles can indeed be used for Lagrange variation. This is done for the gyro with one point fixed.

I calculated the transformation matrix A of the note and its inverse. The order of rotations is different from that in M&T.

Horst

Am 24.02.2017 um 15:11 schrieb EMyrone:

In this note the reference frames used in Note 371(2) are defined. The mass m orbits a mass M situated at the origin. The spherical polar coordinates are defined, and frame (X, Y, Z) is rotated into frame (1, 2, 3) with the matrix of Euler angles as in Eq. (11). The inverse of this matrix can be used to define r1, r2, and r3 in terms of X, Y, Z. In plane polar coordinates the orbit is a conic section with M at one focus as is well known.The planar elliptical orbit does not precess, but using spherical polar coordinates and Eulerian angles the orbit is no longer planar and precesses.

This is a very good idea, many ideas such as this can be developed.

To: EMyrone@aol.com
Sent: 25/02/2017 07:21:27 GMT Standard Time
Subj: PS: Re: Note 371(3) : Definition of Reference Frames

PS: a solution can be to consider a masspoint with 3 internal degrees of freedom, corresponding to the Euler angles. In addition there are 3 degrees of freedom for external motion (cartesian or spherical coordinate system). The internal degrees of freedom represent a spin resp. spin moment. It is not useful to introduce further translational coordinates arbitrarily on the surface of a body or so, because this does not conform to the Lagrange concept. The latter is made for masspoints only.

Horst

Am 24.02.2017 um 19:34 schrieb Horst Eckardt:

Thanks, this clarifies the subject. I think we must be careful in applying Lagrange theory. A mass point in 3D is described by 3 variables that are either [X. Y. Z] or [r1, r2, r3]. The Eulerian angles describe the coordinate transformation between both frames of reference bold [i, j, k] and bold [e1, e2, e3]. The Eulerian angles are the sam for all points [r1,
r2, r3]. So they cannot be subject to variation in the Lagrange mechanism. The degree of freedom must be the same in both frames, otherwise we are not dealing with generalized coordinates and Lagrange theory cannot be applied.

A Lagrange approach in Eulerian coordinates can be made if we describe the motion of a mass point that is described as a unit vector in the [e1, e2, e3] frame. Then [r1, r2, r3] is fixed and the Eulerian angles can indeed be used for Lagrange variation. This is done for the gyro with one point fixed.

I calculated the transformation matrix A of the note and its inverse. The order of rotations is different from that in M&T.

Horst

Am 24.02.2017 um 15:11 schrieb EMyrone:

In this note the reference frames used in Note 371(2) are defined. The mass m orbits a mass M situated at the origin. The spherical polar coordinates are defined, and frame (X, Y, Z) is rotated into frame (1, 2, 3) with the matrix of Euler angles as in Eq. (11). The inverse of this matrix can be used to define r1, r2, and r3 in terms of X, Y, Z. In plane polar coordinates the orbit is a conic section with M at one focus as is well known.The planar elliptical orbit does not precess, but using spherical polar coordinates and Eulerian angles the orbit is no longer planar and precesses.

It is a good idea to use the gyro with one point fixed for orbital theory, the mass M is at the fixed point of the gyro, and mass m is separated by a distance r from M. Unlike problem 10.10 of Marion and Thornton, however, the distance r is not constant. I will look in to this and go back to the basics of the derivation of the Euler equation from variational calculus, Marion and Thornton chapter five (Euler 1744). Howevber, I think that all is OK for the following reasons. It is true that the Euler angles relating frame (i, j, k) and (e1, e2, e3) are constants by definition, provided that frame (e1, e2, e3) always has the same orientation with respect to frame (i, j, k) and provided that the two frames are static. Then theta, phi and chi, being constants, cannot be used as variables. I agree about this point. However in Note 371(2), frame (e1, e2, e3) is moving with respect to (i. j. k), and so theta, phi and chi are also moving. This is because the Cartesain frame (i, jk, k) is static by definition, but frame (e1, e2, e3) is dynamic, i.e. e1, e2 and e3 depend on time, but i, j, and k do not depend on time. Similarly in spherical polars, (i, j, k) static, but (e sub r, e sub theta, e sub phi) is time dependent so r, theta and phi all depend on time. The lagrangian (1) of Note 371(2) is true for any definition of v, and Eq. (10) of that note is true for any definition of the spin connection (e.g. plane polar, spherical polar, Eulerian, and any curvilinear coordinate system in three dimensions). So Eqs. (11) to (16) of that note are correct. So it is correct to set up the lagrangian (16) using the Lagrange variables r1(t), r2(t), r3(t), theta(t), phi(t) and chi(t). It could also be set up with plane polar or spherical polar coordinates. We have already correctly solved those problems using the lagrangian method, The fundamental property of the spin connection is to show how the axes themselves move and it is valid to re express the angles of the plane polar and spherical polar coordinates as Eulerian angles. In the orbital problem, the Eulerian angles are all time dependent. So they vary in this sense, and can be used as Lagrange variables. The Euler variable x of chapter five of Marion and Thornton is t, x = t. To sum up, it is true that the Euler angles are constants when viewed as angles defining the orientation of a static (1, 2, 3) with respect to a static (X, Y, Z), but in Eq. (10) of Note 371(2), the components of the spin conenction are defined in terms of time dependent Euler angles, which are therefore Lagrange variables. This fact can be seen from Eqs. (6) to (8) of the note, in which appear phi dot, theta dot, and chi dot. These are in general non zero, i.e. they are all time dependent angular velocities.

To: EMyrone@aol.com
Sent: 24/02/2017 18:35:19 GMT Standard Time
Subj: Re: Note 371(3) : Definition of Reference Frames

Thanks, this clarifies the subject. I think we must be careful in applying Lagrange theory. A mass point in 3D is described by 3 variables that are either [X. Y. Z] or [r1, r2, r3]. The Eulerian angles describe the coordinate transformation between both frames of reference bold [i, j, k] and bold [e1, e2, e3]. The Eulerian angles are the sam for all points [r1, r2, r3]. So they cannot be subject to variation in the Lagrange mechanism. The degree of freedom must be the same in both frames, otherwise we are not dealing with generalized coordinates and Lagrange theory cannot be applied.

A Lagrange approach in Eulerian coordinates can be made if we describe the motion of a mass point that is described as a unit vector in the [e1, e2, e3] frame. Then [r1, r2, r3] is fixed and the Eulerian angles can indeed be used for Lagrange variation. This is done for the gyro with one point fixed.

I calculated the transformation matrix A of the note and its inverse. The order of rotations is different from that in M&T.

Horst

Am 24.02.2017 um 15:11 schrieb EMyrone:

In this note the reference frames used in Note 371(2) are defined. The mass m orbits a mass M situated at the origin. The spherical polar coordinates are defined, and frame (X, Y, Z) is rotated into frame (1, 2, 3) with the matrix of Euler angles as in Eq. (11). The inverse of this matrix can be used to define r1, r2, and r3 in terms of X, Y, Z. In plane polar coordinates the orbit is a conic section with M at one focus as is well known.The planar elliptical orbit does not precess, but using spherical polar coordinates and Eulerian angles the orbit is no longer planar and precesses.

371(3).pdf

The equivalent of 107,750 printed pages was downloaded (392.856 megabytes) from 2,422 downloaded memory files and 506 distinct visits each averaging 3.6 memory pages and 9 minutes, printed pages to hits ratio of 44.49, Top ten referrals total 2,212,224, main spiders Google, MSN and Yahoo. Collected ECE2 1117, Top ten 961, Collected Evans / Morris 759(est), Collected scientometrics 530, F3(Sp) 337, Barddoniaeth 250, Principles of ECE 148, Collected Eckardt / Lindstrom papers 116, Autobiography volumes one and two 107, Collected Proofs 87, Engineering Model 82, UFT88 65, Evans Equations 55, PECE 46, CEFE 42, ECE2 41, UFT311 41, Self charging inverter 28, Llais 20, PLENR 11, UFT313 20, UFT314 19, UFT315 15, UFT316 17, UFT317 28, UFT318 15, UFT319 19, UFT320 15, UFT322 20, UFT323 15, UFT324 16, UFT325 25, UFT326 17, UFT327 16, UFT328 26, UFT329 21, UFT330 13, UFT331 19, UFT332 14, UFT333 19, UFT334 14, UFT335 18, UFT336 14, UFT337 15, UFT338 13, UFT339 12, UFT340 12, UFT341 21, UFT342 11, UFT343 17, UFT344 20, UFT345 14, UFT346 14, UFT347 19, UFT348 14, UFT349 18, UFT351 19, UFT352 27, UFT353 25, UFT354 28, UFT355 16, UFT356 25, UFT357 24, UFT358 18, UFT359 19, UFT360 16, UFT361 13, UFT362 21, UFT363 20, UFT364 23, UFT365 15, UFT366 42, UFT367 41, UFT368 44, UFT369 42, UFT370 24 to date in February 2017. University of Quebec Trois Rivieres UFT366 to UFT370; Leibniz Institute for Neurobiology Magdeburg LCR Resonant; Spanish Ministry for Employment and Social Security F3(Sp); Chemistry University of Ovieda Spain My page, list of most prolific scientists, CV; Samsung South Korea Levitron; Mexican National Polytechnic Institute Center for Research and Advanced Studies (Cinvestav) UFT199(Sp); Swedish Digital Freedom and Rights Association general; University of Edinburgh extensive, including Wales and Welsh affairs, UFT369, UFT370, PECE; University of Leeds experimental advantages over the standard model. Intense interest all sectors, updated usage file attached for February 2017.

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In this note the reference frames used in Note 371(2) are defined. The mass m orbits a mass M situated at the origin. The spherical polar coordinates are defined, and frame (X, Y, Z) is rotated into frame (1, 2, 3) with the matrix of Euler angles as in Eq. (11). The inverse of this matrix can be used to define r1, r2, and r3 in terms of X, Y, Z. In plane polar coordinates the orbit is a conic section with M at one focus as is well known.The planar elliptical orbit does not precess, but using spherical polar coordinates and Eulerian angles the orbit is no longer planar and precesses.

a371stpapernotes3.pdf

This is the usual orbital problem of m attracted by M according to the usual Hooke / Newton inverse square law of force, the concept of force having been proposed by Kepler (Koestler, “The Sleepwalkers” online). The frame (1, 2, 3) is a generalization of the usual frame (plane polar or spherical polar). So in a planar orbit (r, 1, 2, 3) reduces to (r, phi) of the plane polar coordinates, in a three dimensional orbit (r, 1, 2, 3) reduces to (r, theta, phi) of the spherical polar coordinates). So the Eulerian angles are used to define the spin connection matrix in Eq. (10), with omega sub i, i = 1, 2, 3 defined by Eqs. (6) to (8). Therefore the angular velocity expressed in terms of spherical polar coordinates has been re expressed in terms of Eulerian angles as in Note 370(9) and similarly for the spin connection. The Eulerian angles define the rotation from the inertial frame (X, Y, Z) to frame (1, 2, 3). The inertial frame contains only the Newtonian force as you know (no centrifugal or Coriolis forces). In the spherical polar coordinate system r bold = r e sub r, where e sub r is the radial unit vector. In the (1, 2, 3) frame the same r bold is defined as

r bold = r sub 1 e sub 1 + r sub 2 e sub 2 + r sub 3 e sub 3

= r e sub r

= X i + Y j + Z k

Here r bold is the vector joining m and M. So to sum up, everything is expressed in frame (1, 2, 3), giving a huge amount of new information which was of course intractable in the eighteenth century.

To: EMyrone@aol.com
Sent: 23/02/2017 10:44:59 GMT Standard Time
Subj: Re: 371(2): Orbital Theory in Terms of Euler Angles

I am not sure if I understand correctly the physical problem to be solved. You have always to discern between the lab and rotating frames. The Lagrangians (1) and (16) are for the rotating frame. So what is r_1, r_2, r_2 in the rotating frame? So far we have considered situations with centre in the rigid body and with a point outside with fixed distance to the centre of the rigid body (gyro with one point fixed). As soon as we relate this to anohter centre of gravity, we obtain additional vectors from the centre of gravity to the centre of rotating frame (1,2,3). I am not sure if eq.(10) is right in this context. We need the complete coordinate transformation to the lab frame and then can define the kinetic energy plus the roational part which is additionally required because of the rigid body.

Horst

Am 23.02.2017 um 11:02 schrieb EMyrone:

It is shown straightforwardly in this note that the orbits of a mass m around a mass M in the frame (1, 2, 3) of the Euler angles can be found by solving simultaneously six Euler Lagrange equaions in six Lagrange variables. The problem is far too complicated to be solved by hand but can be solved numerically using Maxima to give twenty one new types of orbit described on page 4. There are nutations and precessions in the Euler angles phi, theta and chi. The usual assumption made in orbital theory is that the orbit is planar. Note carefully that the inverse square law is still being used for the force of attraction between m and M, but the Eulerian orbits contain far more information than the theory of planar orbits. The inverse square law does not give precession of a planar orbit as is well known. However the same inverse square law gives rise to many types of nutation and precession in the twenty one types of orbit described by the Euler angles. This isa completely new discovery, and it should be noted carefully that it is based on classical rotational dynamics found in any good textbook. It could have been defined in the eighteenth century, but orbital theory became ossified in planar orbits. This theory is part of ECE2 generally covariant unified field theory because all of rotational dynamics are described by a spin connection of Cartan geometry. Eq. (10) of this note is a special case of the definition of the Cartan covariant derivative. This is another major discovery. An art gallery full of graphics can be prepared based on these results, of greatest interest are plots of the twenty one types of orbit, looking for precessions of a point in the orbit such as the perihelion.

This plan looks full of interest, giving orbits r(theta), r (phi) and r (beta). There is a lot of interest already in the latest papers as can be seen from the early morning reports.

To: EMyrone@aol.com
Sent: 23/02/2017 10:13:07 GMT Standard Time
Subj: Re: Checking Note 371(1), corrigendum Eq. (6).

Ok, the motion in coordinates of r, theta, phi could be computed numerically from eqs.(3-5), and beta could be determined from the additional diff. equation (6). This then gives the constant of motion L which is not known a priori, if we start the solution from the initial conditions. The corresponding argument holds for L_Z.

Horst

Am 23.02.2017 um 09:21 schrieb EMyrone:

Many thanks again. This is just a typo, the equation should be the same as Eq. (25) of UFT270. The rest of the note is the same. The great advantage of UFT371 over UFT270 is that Maxima is able to solve all the relevant equations numerically in UFT371, so the laborious hand calculations in UFT270 no longer have to be done. A close control over the use of the computer is still needed of course, but an array of new possibilities opens up. I will proceed now to develop the same problem in terms of the Eulerian angles. ECE2 relativity is still needed of course in other situations, but it would be interesting to see whether classical rotational dynamics gives planetary precessions. That would mean that one of the most famous experiments of Einsteinian general relativity EGR, the precession of the perihelion would be refuted. We have refuted EGR in many ways, and there have been no valid objections to these refutations. The lagrangian (1) of Note 371(1) gives precessions in the angles of the spherical polar coordinates. When Eulerian angles are used, more precessions appear on the classical level. Furthermore, we now know clearly that classical rotational dynamcis are governed by a well defined spin connection of Cartan geometry. Finally, all these calculations can be quantized. Again, Maxima removes all the laborious calculations.

To: EMyrone
Sent: 22/02/2017 13:48:28 GMT Standard Time
Subj: Re: Note 371(1): Precession of the Perihelion on the Classical Level

Anything seems to go wrong with beta. Inserting beta dot squared from (6) does not give (1), there are additional terms then.

Horst

Am 21.02.2017 um 12:31 schrieb EMyrone:

This note looks afresh at the precession of the perihelion by setting up the classical lagrangian (1) and solving Eqs. (3), (4), (5), (9) and (13) simultaneously for the orbit r = alpha / (1 + epsilon cos beta) where beta is defined in Eq. (6). This method is a development of one used originally in UFT270. The power of the Maxima program now allows the relevant equations to be solved for beta in terms of the angles theta and phi if the spherical polar coordinates system. There are precessions in theta and phi. The precession of the perihelion is usually thought of as a precession of phi in a planar orbit, using the incorrect Einsteinian general relativity. In the UFT papers ECE2 relativity has been used to describe the precession. However it may be that it can be described on the classical level with the use of spherical polar coordinates. If this supposition is true, and if Eq. (8) is a precessing ellipse, then other precessions can also be developed in this way. The theory can also be developed with the Eulerian angles. In general there are precessions in theta and phi. There is no reason why an orbit should be planar. In general it must be described by a three dimensional theory.

The equivalent of 91,258 printed pages was downloaded (332.726 megabytes) from 2.752 downloaded memory files (hits) and 506 distinct visits each averaging 4.1 memory pages and 8 minutes, printed pages to hits ratio of 33.17, top ten referrals total 2,211,875, main spiders google, MSN and Yahoo. Collected ECE2 1076, Top ten 947, Evans / Morris 726(est), Collected scientometrics 467, F3(Sp) 312, Barddoniaeth 208, Principles of ECE 143, Eckardt / Lindstrom 113(est), Autobiography volumes one and two 95, Collected proofs 84, Engineering Model 80, UFT88 63, Evans Equations 53, PECE 43, UFT311 41, CEFE 41, ECE2 36, Self charging inverter 28, UFT321 22, Llais 19, PLENR 11, UFT313 20, UFT314 18, UFT315 14, UFT316 15, UFT317 28, UFT318 15, UFT319 19, UFT320 15, UFT322 19, UFT323 15, UFT324 16, UFT325 23, UFT326 16, UFT327 16, UFT328 25, UFT329 21, UFT330 13, UFT331 19, UFT332 14, UFT333 19, UFT334 14, UFT335 18, UFT336 12, UFT337 15, UFT338 11, UFT339 11, UFT340 12, UFT341 21, UFT342 11, UFT343 17, UFT344 20, UFT345 14, UFT346 13, UFT347 17, UFT348 14, UFT349 18, UFT351 19, UFT352 25, UFT353 25, UFT354 28, UFT355 15, UFT356 24, UFT357 23, UFT358 17, UFT359 18, UFT360 16, UFT361 13, UFT362 20, UFT363 20, UFT364 23, UFT365 14, UFT366 37, UFT367 39, UFT368 43, UFT369 39, UFT370 21 to date in February 2017. Johann Radon Institute for Computational and Applied Mathematics Johannes Kepler University Linz UFT149; Deusu search engine filtered statistics; Cornell University UFT142; University of California Los Angeles UFT146; Laboratory for the Structure, Properties and Modelling of Solids Ecole Centrale Paris (Grande Ecole) UFT331; Dr Pip Nicholas Davies, University College of Wales Aberystwyth Definitive Proof 1. Intense interest all sectors, updated usage file attached for February 2017.

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