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 AttachmentMany 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