Yes these are the four notes to date, I have just started work on three dimensional galactic dynamics.
In a message dated 01/09/2014 10:52:41 GMT Daylight Time, firstname.lastname@example.org writes:
Is there also a note 270(2)?
My internet has been checked today morning, obviously someone had temporarily interrupted the line in the house, seems to work again now.
EMyrone@aol.com hat am 1. September 2014 um 10:59 geschrieben:
The condition for planar orbits is Eq. (17) or (27), the equation which results from the assumption that the constants of integration of Eq. (15) or (16) are zero. So this is the reason why planar orbits are planar. If one time dependent constant of integration is chosen as in Eq. (23), then the orbit is three dimensional and all previous work based on Eq. (24) remains valid. So Eq. (24) is the one to use in general. The three dimensional Binet equation is Eq. (26). The three dimensional orbit depends in general on the function (27) in which beta is a function of phi and theta. Computer algebra can be used to re express Eq. (27) in terms of phi and theta. I can also do this by hand using the chain rule of differential calculus. There are many interesting types of three dimensional orbits in galaxies, and they are all described by Eq. (27). All of this analysis is based on the spherical polar coordinates and lagrangians, so no one can object to the results, especially as they are checked many times by hand and computer algebra.