Relativistic Kepler motion

H0 is the Hamiltonian function of the non-relativistic Kepler motion, which we regard as the unperturbed motion, and Hx is a perturbation function. In order to find the influence of this perturbation on the Kepler motion, we have to average Hx over the unperturbed motion. If we express the sum of the squares of the momenta occurring in Hx with the help of the equation for W0, we obtain... 

An example will at once make clear what is meant. Let us take the relativistic Kepler motion, or, in other words, motion in an ellipse with a perihelion rotation. In general, the path fills a circular ring and, therefore, a two-dimensional region, densely everywhere. The boundaries for the libration of the radius vector are here concentric circles. 

The expansions of the cartesian co-ordinates as functions of the angle variables (to be calculated from (26), 22) must now be introduced, to provide a starting-point for the calculation of the perturbations. In this connection, however, there is one point to be borne in mind. In the unperturbed Kepler motion (without taking account of the variation in mass) only Jx is fixed by the quantum theory, whilst J2, i.e. the eccentricity, remains arbitrary in the relativistic Kepler motion, J2 is also to be quantised and, for a one-quantum orbit, J2=J1=A. We shall not take account quantitatively of the relativistic variation of mass, but we shall assume that the initial orbit of each electron is circular with limiting degeneration J1=A,... 

Let us resolve this function into H0 and H1, where H0 is the Hamiltonian function of the (non-relativistic) Kepler motion of the inner electron and Hx the remaining part of the above expression. 

Atomic Structure and Chemical Properties 31. The Actual Quantum Numbers op the Optical Terms 32. The Building Up op the Periodic System op the Elements. 33. The Relativistic Kepler Motion. ... 

In Sect. 1.4, we will demonstrate the validity of the method by analysing the relativistic Kepler problem by computing the perihelion motion of the planet Mercury, followed by Sect. 1.5, displaying the explicit connection between the Schwarzschild singularity and Gddel s theorem. The final conclusion summarises the modus operandi and its subsequent consequences. 

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