## 362(2a): Orbital Dynamics in the Elliptical Polar Coordinate System

This note defines the elliptical polar coordinate system in which the contradiction between Eqs. (13) and (14) of orbital theory in the plane polar system is removed. The r, v and a vectors of the elliptical polar system are defined by Eqs. (15), (17) and (18). It is checked that Eq. (17) gives the Newtonian result for a circular orbit, but for the conic section orbits several new velocities and accelerations appear, defined by Eqs. (52) and (53), and not inferred by Coriolis or anyone else up to now. So the velocities and accelerations of dynamics are different for the plane and elliptical polar systems. The latter is the one that should be used for a conic section orbit. Computer algebra can be used to work out Eqs. (17) and (18) in which theta is a function of t. Theta can be transformed to r using Eqs. (19) and (20). Computer algebra is needed at all stages, because the calculations get complicated, even though te basic definitions (1) to (5) are well known and simple. The unit vectors of the elliptical polar system are given by Eqs. (35) and (36). These should be checked by computer algebra. As usual my hand calculations are always checked by co author Dr Horst Eckardt using the computer in order to eliminate human error. It follows after some algebra that in the elliptical polar system, the familiar Eq. (51) of the plane polar system no longer holds.The algebra is straightfoward but a little complicated, so should be checked by computer. The fact that Eq. (51) no longer holds leads to several new velocities given by Eq. (52) and accelerations, given by Eq. (53). The computer should be used to find whether these reduce to the familiar eqs. (11) and (12). If not, entirely new dynamics would have been dsicovered. This work looks complicated, and the concepts are not easy for the non specialist, but the basic idea is simple, to use a coordinate system in which the self contradiction between Eqs. (13) and (14) vanishes, and then to evaluate the velocities and accelerations in this new coordinate system. Effectively, use is made of a coordinate system that is self consistent with the observed orbit, the conic section (5) exemplified by the ellipse. On a more refined level a precessing elliptical coordinate system can be used. The next stage is to find the spin conenction matrix for the elliptical polar coordianate system and genrealize the calculation using the convective derivative, which must be evaluated in the elliptical polar system. This elliptical polar system is completely new. For background reading I recommend “Vector Analysis Problem Solver”.

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