This is full of interest, the results are completely original as far as I know. They go well beyond UFT270 to UFT276 due to the use of numerical methods. They show that spherical polar coordinates produce planar orbits which can tilt, so the plane of the orbit can tilt, depending on initial conditions. This is exactly what is observed in the solar system, where some orbits are tilted. Up to now there has been no satisfactory explanation of why the orbits of some planets are tilted with respect to the orbits of other planets. The orbits are planar even though spherical polar coordinates are used as proper Lagrange variables. The introduction of angle dependent potentials is full of interest, both in this context and in the context of Milankowitch cycles. In this context a very interesting type of orbit is obtained, similar to the one pointed out by Norman Page from the literature. As Horst mentions it may evolve into a Moebius strip. In the context of Milankowitch cycles an angular dependent potential was proposed in order to obtain an interaction between the orbit of the earth and its own nutations and precessions. Finally, spherical polar coordinates do not appear to give orbital precession, so relativity is indeed needed for that effect, for example UFT328. In UFT372 I will proceed to the relativistic Lagrangian theory, both of orbits and quantum mechanics. These results show the great power of numerical methods and graphics even though we are using only a desktop. A mainframe and supercomputer could easily crunch out Lagrangian quantum mechanics.
Sent: 04/03/2017 20:36:46 GMT Standard Time
Subj: Section 3 of paper 371
I am sending over section 3. Angular-dependent potentials can produce 3D
curves with periodicity 4 pi for example.This could be the basis for a
It seems to be difficult to find a modification of the gravitational
potential that produces precessing ellipses. This seems rather to be
effect of modified dynamics as we found for the relativistic Kepler problem.