Precessing Orbits with Cartesian Coordinates

I like this work, it shows clearly that the addition of a velocity field gives a precessing orbit, a remarkable result. I have a few comments.

1) If the field terms v sub fX and v sub fY are zero there is no centrifugal force, in which case there is no orbit. This can be checked with computer algebra.
2) It would be interesting to find under what conditions a static ellipse is obtained. In that case the fields are acting as if they were a centrifugal force.
3) It is interesting to graph the kinds of orbit that are given by different fields.
4) Compare the precessing orbit with experimental data.

The basic idea is that spacetime or ather is a fluid.

To: EMyrone@aol.com
Sent: 06/04/2017 09:31:37 GMT Daylight Time
Subj: Re: A direct approach to mass point dynamics in a fluid

I programmed this model, this time in cartesian coordinates so that velocities of the fluid can be defined more easily. I used the Lagrange method with Lagrangian

A general velocity field (vfx, vfy) has been added in the kinetic energy terms. Then the most general Euler-Lagrange equations then are:

For a constant vf there is no change in the equations of a masspoint without fluid velocity, that means such a case is contained in appropriate initial conditions. By defining any form of vf one can obtain various orbit types as I found in the ECE2 fluid dynamics case. For example

gives a rosette which is a precessing ellipse with large precession angle, see Fig. 6. The precession can be minimized by a0 –> 0.

Horst

Am 05.04.2017 um 09:11 schrieb Horst Eckardt:

What about the following approach (in cartesian coordinates):
Given: a velocity field bold vf(X,Y,Z,t)
Coordinates of mass point: X,Y,Z
velocity of mass point:

bold v = (v_X + vf_X, v_Y + vf_Y, vZ + vf_Z)

with v_X = X dot, etc.

The Lagrange equations can be obtained from the above equation for bold v. It has to be checked if this is a valid precedure because in Lagrange theory normally the coordinates are transformed and not the velocities. As an alternative, one could certainly solve Newton’s equations directly.
This model does not include viscosity or other additional effects.

Horst

Simplex-XY.pdf