OK many thanks, this Runge Kutta method is a very powerful Maxima algorithm, and can be applied now to give a large amount of new results for many problems of ECE2 theory. I will write out the equations of most interest. For spherical orbits it will be very interesting to plot the true orbit r = alpha / (1 + epsilon cos beta) obtained from the Hooke Newton inverse square law using spherical polar coordinates. No modelling assumption is made other than the usual inverse square law.
Sent: 27/02/2017 17:38:36 GMT Standard Time
Subj: Re: 371(4): Scheme of Computation for the 3D Orbit
Basically thre are two methods of solving the central motion in a spherical polar coordinate system: Either the full set of Lagrange equations without constants of motion or by predefining L and L_Z and solving (19-22). I will try the latter. These equations are only of first order for beta, theta, phi so they reduce the number of Hamilton equations used for the Runge-Kutta solution method.
Am 26.02.2017 um 14:19 schrieb EMyrone:
This note gives the scheme of computation that gives all information about a spherical orbit governed by an inverse square force law of attraction. Maxima can now be used to compute any type of information needed, notably the orbits r(beta), r(theta) and r(phi). Some of the analytical results from UFT270 to UFT276 on three dimensional orbits are reviewed. The planar orbit is recovered in the limit theta goes to pi / 2, theta dot goes to zero. The planar orbit is given by Eq. (33), and does not precess for an inverse square law as is well known. In three dimensions however it precesses in various ways. There is a typo in Eq. (31), which should read theta goes to pi / 2. We are now ready to use Maxima for the lagrangian in ECE2 relativity, both for planar and three dimensional orbits.