270(3) : The Condition for Planar Orbits

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 lagrangian equations, so no one can object to the results, especially as they are checked many times by hand and computer algebra.

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