Agreed about the factor two. The R sub r function is as defined in UFT363 as the radial component of the position of a fluid element. The fluid in this case is spacetime itself. If the fluid element reduces to a point particle, partial R sub r / partial r vanishes, because R sub r = R sub r( r(t), theta(t) t), whereas r = r(t), i.e. in classical orbital theory r is a function only of t bu definition and is not a function of r(t) or of phi(t). Agreed about Egs (37) to (40) being not yet a well defined set. Another equation is needed, and agreed also about the final point.
Sent: 01/04/2017 18:46:44 GMT Daylight Time
Subj: Re: 374(3): Equations for the Precessing Orbit of Fluid Gravitation
There seems a factor of 2 be missining in eqs. 14 and 28. Let me give three comments on this note:
Concerning solvability of eqs. 37-40, see my comments for the previous note. I see that eq.(40) is derived correctly from (39), but (37,38,40) seems not to be a meaningful equation set, and the numerical solution with a model function R_r(r) showed that the angular momentum is not conserverd.
Concerning interpretation of R_r, if
x = 1 + partial R_r / partial r,
then R_r is constant in the case x –> 1. If R_r were the radius function, we had in this case
R_r = r
and x –> 2. The function R_r seems rather to be the deviation of radial motion from the coordinate r of a fluid element.
The equation system (37,38,40) could be consistent, if only a time dependence of x is considered. You computed this in note 374(4). I will do an equation check and numerical solution if possible.
Am 30.03.2017 um 15:38 schrieb EMyrone:
In the first instance, Eqs. (27) to (29) can be solved numerically using Maxima to check that the method gives the correct orbit (18). Then the algorithm can be modified to solve Eqs. (37), (38) and (40 numerically using a model for the function x defined in Eq. (31). Finally Eq. (44) can be added if the fluid is assumed to be incompressible, so both the orbit and x can be found. The caveat of this note explains why the note slightly corrects the equations of UFT363. The lagrangian method of UFT363 gives Eq. (47), which is different from the correct Eq. (37). The reason is that the kinetic energy of fluid gravitation, Eq. (48), is not in the required format (49) demanded by the Hamilton Principle of Least Action. The kinetic energy must be T(r bold dot, r bold). Sometimes it is simpler and clearer to derive results without the Lagrange method using both the Lagrange and Hamilton equations of motion.