Alpha Institute for Advanced Studies

In this final version the successive approximation method produces the solution (11) in which the eccentricity is kept constant. It is the sum of a particular integral and a complementary function. The constants C1 and C2 of the complementary function are arbitrary and for simplicity can be chosen to vanish, but in general they are non-zero and the choice of these constants affects the orbit. This is true even if the spin connection vanishes, i.e. they are also present in Newtonian dynamics when the Binet equation is used. So the Newtonian method does not uniquely determine an orbit. A choice of zero C1 and C2 gives the relatively simple function (7). This can be compared with the orbit given by numerical integration of Eq. (1) of Note 403(3). The spin connection is defined by vacuum fluctuations which produce the precession of any orbit, and indeed any cosmological precession.

a403rdpapernotes4.pdf

Many thanks! I will refine the method and solution in further notes to UFT403, and Horst can add the numerical solution. In the present solution there is a singularity at cos phi = – 1 , so I wish to find another solution without this singularity. The basic method however seems to be fine. The general solution is the sum of a particular integral and a complementary function, as usual in the theory of differential equations.
Myron Evans <myronevans123>

Fantastic Myron. This is relentless progess. A privelege to witness it.

Subject: 403(3): Analytical Solution of the ECE2 Force Equation of Orbits

The solution is given in Eq. (26) and produces the same precession (40) as in Note 403(2), given the same approximations. The solution (26) is the sum of a particular integral and a complementary function. It also appears to be new to mathematics, it does not appear in the Wolfram online integrator for example. Precessions of all types can be described by equation (26), which calculates the precession in terms of only one spin connection component and which shows that the origin of precession of all types is the vacuum. The analytical solution is very useful because numerical solutions can be checked against the analytical solutions. The spin connection can be related to vacuum fluctuations as in recent UFT papers, and can also be related to the relativistic Newton equation.

Many thanks again!

The solution is given in Eq. (26) and produces the same precession (40) as in Note 403(2), given the same approximations. The solution (26) is the sum of a particular integral and a complementary function. It also appears to be new to mathematics, it does not appear in the Wolfram online integrator for example. Precessions of all types can be described by equation (26), which calculates the precession in terms of only one spin connection component and which shows that the origin of precession of all types is the vacuum. The analytical solution is very useful because numerical solutions can be checked against the analytical solutions. The spin connection can be related to vacuum fluctuations as in recent UFT papers, and can also be related to the relativistic Newton equation.

a403rdpapernotes3.pdf

The ECE2 force equation gives the very simple result (35) in the same approximation as used in Note 403(1). There does not seem to be an analytical solution to Eq. (20), but this approximate method shows that it produces precession of the half right latitude. Any experimentally observed orbital precession can be explained exactly by adjusting the spin connection, so any precession can be explained exactly by the vacuum in a very simple way. Maxima can be used to find the numerical solution of Eq. (20). By Ockham’s Razor this theory is preferred for small precessions to all more complicated theories. In the next not I plan to apply the successive approximation method to Eq. (20), but the best solution is the Maxima solution, using its numerical integration routines.Finally we have traced the origin of teh spin onnection to vacuum fluctuations of the type used in Lamb shift theory, which also gives a precise match with experiment.

a403rdpapernotes2.pdf