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OK many thanks. In Eq. (18) the lower case v should have been a capital V.

362(5): Convective Derivative of the General Vector, Application to Orbits

I have a problem with eqs.(17-21) which should hold for arbitrary vectors V. The middle term at the RHS in (21) only depends on the velocity v. This seems not be meaningvul if V is not a velocity. I think this should be:

DV/dt = partial V / partila t + (v*nabla) V .

There is everywhere a vector V to be used at the RHS of the operators. In the operator v*nabla itself the velocity appears. This gives different results for (22).

Horst

Am 24.11.2016 um 13:15 schrieb EMyrone:

The convective derivative of any vector field V(t, r(t), theta(t)) is given by Eq. (7) and involves two Cartan derivatives as shown. The result is applied to the Coriolis velocity in Eq. (22) and the Coriolis accelerations in Eq. (23). The components of the orbital Coriolis velocity are changed to Eqs. (35) and (36) when the orbiting object of mass m is considered to move through a fluid spacetime or aether instead of as a point particle as in the usual orbital theory. The observable square of the orbital velocity is changed to Eq. (38), and the Newtonian result (39) is changed by the presence of the spin connection (26) of fluid dynamics and Cartan geometry. By suitable choice of spin connection components it should be possible to produce a precessing orbit, and also the hyperbolic spiral orbit of a whirlpool galaxy. The kinetic energy, hamiltonian and lagrangian are changed by the assumption of a fluid spacetime, aether or vacuum. Computer algebra could be used to model a precessing orbit and hyperbolic spiral orbit with choice of spin connection components. At this point I will write up Sections 1 and 2 of UFT362 and pencil in Section 3 for co author Horst Eckardt as usaul.

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The convective derivative of any vector field V(t, r(t), theta(t)) is given by Eq. (7) and involves two Cartan derivatives as shown. The result is applied to the Coriolis velocity in Eq. (22) and the Coriolis accelerations in Eq. (23). The components of the orbital Coriolis velocity are changed to Eqs. (35) and (36) when the orbiting object of mass m is considered to move through a fluid spacetime or aether instead of as a point particle as in the usual orbital theory. The observable square of the orbital velocity is changed to Eq. (38), and the Newtonian result (39) is changed by the presence of the spin connection (26) of fluid dynamics and Cartan geometry. By suitable choice of spin connection components it should be possible to produce a precessing orbit, and also the hyperbolic spiral orbit of a whirlpool galaxy. The kinetic energy, hamiltonian and lagrangian are changed by the assumption of a fluid spacetime, aether or vacuum. Computer algebra could be used to model a precessing orbit and hyperbolic spiral orbit with choice of spin connection components. At this point I will write up Sections 1 and 2 of UFT362 and pencil in Section 3 for co author Horst Eckardt as usaul.

a362ndpapernotes5.pdf

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Many thanks to Gareth Evans. There are many combinations and permutations possible now, so it is a matter of selecting the most incisive. The third book is pencilled in as “The Evans / Morris Effects” and planned to be four authored, Trevor and yourself and Horst and myself. Improvements in printing technology will enable the inclusion of many of your colour photographs. You may like to discuss the choice of photographs with Trevor Morris and in the meantime review the Evans Morris papers on www.aias.us and www.upitec.org.

To: EMyrone@aol.com
Sent: 23/11/2016 14:47:17 GMT Standard Time
Subj: Re: 362(4): The Coriolis velocity as a Cartan derivative

Your reference to the weak and strong nuclear fields are also very important of course and will clear up loads of uncertainty as well as probably suggesting new areas of work and useful applications.

Sent from my Samsung device

This note derives the well known Coriolis velocity (10) of the plane polar coordinate system as the generally covariant Cartan derivative (1) of ECE2 unified field theory. The Coriolis velocity includes the well known orbital velocity v = omega r of a circular orbit. Note carefully that this result is no longer true in an elliptical orbit. As shown in Note 362(2a), and computer algebra by co author Horst Eckardt, the elliptical polar system is needed for self consistency. The elliptical polar coordinates system is expected to produce new fundamental velocities and accelerations of classical dynamics, this is pencilled in for UFT363. A remarkable result of computer algebra is that Eq. (12) is also true in the elliptical polar system. The general vector field V(t, r, theta) in plane polar coordinates is given by Eq. (17), and its time derivative is the Lagrange derivative, Eq. (18). The spin connection from Eq. (18) will be given in the next note. In classical dynamics, V = V(t). Therefore if an orbit is observed to deviate from the result given by V = V(t), then it is logical to assume that the orbit must be analyzed by V = V(t, r, theta) within the overall structure of ECE2 generally covariant unified field theory, in which fluid dynamics, dynamics, gravitation and electrodynamics are unified by Cartan geometry. It is probable that this unification already encompasses the weak and strong nuclear forces, but this needs to be proven rigorously. UFT225 showed that conventional electroweak theory is a fiasco from which no conclusions can be drawn.

a362ndpapernotes4.pdf

Now that the blog is fixed, these internationally known blog dialogues can be resumed and archived in real time on the wayback machine for the readership (www.archive.org). This looks to be very interesting work and can be the basis of UFT362 along with Notes 362(1) and 362(3) and one or two more notes that I intend to prepare. The UFT363 can be dedicated to the plane elliptical coordinates, when a new discovery in vector analysis appeared out of the blue. All of the work described below is full of interest. Eq. (23), Notes 361(3) gives the complete convective derivative in cylindrical coordinates. It is also given in Cartesian, spherical polar and general curvilinear coordinates on www.pleasemakeanote.blospot.co.uk. Marion and Thornton give the usual acceleration a(t) in spherical polar coordinates as well as cylindrical and Cartesian coordinates. It looks as if the change from plane polar to elliptical polar coordinates produces new accelerations (a(t)) and velocities (v(t)) in classical dynamics. There is a snowstorm of complexity in elliptical polar coordinates, which simplifies tremendously to r bold = r e sub r bold as shown by Horst’s protocol using Maxima. I can take it from there to look for new velocities and accelerations due to the self consistent and rotating elliptical polar frame. These intricate calculations have been be done by hand, and they result in a new discovery in vector analysis. In 1835, Coriolis used a plane polar system, although his mathematical language was probably much different.

To: EMyrone@aol.com
Sent: 22/11/2016 09:44:07 GMT Standard Time
Subj: Re: Discussion of Attachment

Many thaks, you anticipated the answer to my question I would have asked next: Do we have to strictly discern between dynamics of mass points and fluid/continuous field dynamics? So the calculations in my note seem to be right and the additional acceleration terms a_1 can further be simplified in general. I already did some plots and one can see the differences between point and fluid dynamics in the case of the Torkado. This seems to be an appropriate example because it is mostly displayed as a flow field. Then the orbit I selected is a streamline of it. I eliminated all time dependencies by assuming angular momentum conservation in Z direction. Then the angular velocity can be expressed as

theta dot = L0^2 / (m r^2)

with r(theta). This is certainly an approach taken from mass point dynamics but hopefully not too wrong here. I will send over the next version of my note shortly.

PS: do we have a special expression for the fluid dynamics acceleration in cylindrical coordinates, with Z terms?

Horst

Am 22.11.2016 um 10:25 schrieb EMyrone:

Many thanks again. The answer is that the conventional acceleration of classical dynamics is based on the assumption v = v(t). So the convective derivative reduces to the ordinary time derivative because v is a function only of t. So Eq. (2) of Note 361(3) reduces to:

Dv / dt = dv / dt = d(r dot e sub r + r theta dot e sub theta) / dt = a

However, if v = v(t, r(t), theta(t)) then the chain rule of differentiation must be used as in Eq. (2) of Note 361(3), and the spin connection is changed to Eq. (1) of note 362(3). In conventional dynamics and orbital theory, v is not a function of r(t), and v is not a function of theta(t). In other words, in classical dynamics, v is not the velocity field of a fluid, it is the velocity of a particle. Observation in classical dynamics seems to confirm this assumption to some currently unknown degree of approximation. In fluid dynamics however the extra accelerations are present and have been observed through the Navier Stokes equation, vorticity equation and so on. So there are eddies, turbulence, baroclinic forces and so on. In classical dynamics v is the velocity of a particle, in fluid dynamics it is the velocity field of a fluid. If for some reason v = v(t, r(t), theta(t)) in classical dynamics the usual acceleration will be changed. If spacetime, the vacuum or ether is a fluid, these extra accelerations should become observable in dynamics and astronomy. That is the prediction of ECE2.

To: Emyrone
Sent: 21/11/2016 20:20:04 GMT Standard Time
Subj: Fwd: Problem with acceleration a1 ?

I recalculated the terms as described in the attachment. Is there an
error? The acceleration a is massively altered by a_1.

Horst

Following the WordPress fiasco I will repost a few messages, in particular discussions with Horst Eckardt on notes for UFT362.