I would say that the total derivative of r with respect to phi is given by Eq. (10), so this could be used to simplify Eq. (15). For example phi dot partial r / partial phi becomes r dot. A distinction has to be made between coordinates and functions. In the plane polar coordinate system (r, phi), r and phi are independent, so partial r / partial phi is zero for coordinates. If we are dealing with functions, then r = alpha / (1 + eps cos phi) for a static ellipse, and dr / dphi is not zero for functions. So I suggest using Eq. (10) to simplify Eq. (15), and I agree about the use of the chain rule. Eq. (15) is for functions.
Sent: 01/04/2017 20:22:10 GMT Daylight Time
Subj: Re: 374(4): Time Dependent x and Onset of Turbulence
Concerning eq. (15): Shoudn’t the partial derivatives vanish? For example the orbital derivative is
but what is partial r / partial phi? r depends implicitly on phi but don’t the partial derivatives only relate to explicit dependencies?
As an alternative, we could transform all derivatives to time derivatives by the chain rule, for example:
partial phi dot / partial phi = partial phi dot / partial t * partial t / partial phi = phi dot dot / phi dot
(identifying d phi / dt by partial phi / partial t).
Am 01.04.2017 um 13:48 schrieb EMyrone:
This note derives the equations (7) to (9) for a time dependent x factor, with x defined in terms of the position field R sub r of UFT363. These equations can be solved for the orbit in terms of x and x dot. In the first instance these can be used as input parameters. It is already known from computation (UFT363) that a constant x gives a precessing orbit, a major discovery. In order to try to solve for x and x dot the assumption can be made of an inviscid, incompressible fluid spacetime governed by Eq. (14). This gives the additional equation (15) which can be solved simultaneously with Eqs. (7) to (9). The conservation of the angular momentum of fluid spacetime gives the vorticity equation (19) in terms of the Reynolds number R of fluid spacetime. At a particular Reynolds number the spacetime become turbulent, and the turbulence will affect the precessing orbit.