Kepler problem

Sommerfeld, A. and Welker, H. 1938. Artificial limiting conditions in the Kepler problem. Ann. Phys. 32 56-65. 

In his work on the wave equation of the Kepler problem in momentum space (ZS. f. Phys. 74, 216, 1932), E. Hellras has derived a differential equation [Equations (9g) and (10b) in his article] which—after a simple transformation — can be understood as the differential equation of the four-dimensional spherical harmonics in stereographic projection. [With the gracious approval of E. Helleras, we correct the following misprints in his article the number E that appears in the last term of his equations (9f) and (9g) should be multiplied by 4.]... 

Kepler problem the transformation being discussed yields a complete... 

The most important problem that has been solved in detail by using action-angle variables is the Kepler problem of planetary motion. The details of the analysis are not important in the present context, but the form of the Hamiltonian for rotation in a central potential V r) = —k/r, obtained as... 

In quantum mechanical context this invariant of the Kepler problem is often referred to as a Runge-Lenz-Pauli vector. In classical mechanics context it is also sometimes referred to as a Laplace-Runge-Lenz vector. 

It is a constant of the motion for the classical Kepler problem (Saletan and Cromer, 1971). The magnitude of X is proportional to the eccentricity of the orbit and X points in the direction of the major axis. Using the correspondence principle Pauli first showed that its quantum mechanical analog,... 

XVIII. Solution of Schrodinger s Equation for the Kepler Problem... 

We first restrict ourselves to the two-dimensional case and take advantage of the fact that Levi-Civita s regularizing transformation (Levi-Civita 1920) has the agreeable property of transforming perturbed Kepler problems into perturbed harmonic oscillators, i.e. into perturbed linear problems. For a recent account of regularization theory see the article (Celletti 2002) and other contributions in the same volume. 

The procedure of Section 2 for regularizing the planar case now carries over almost identically to the spatial case care must be taken to preserve the order of the factors in quaternion products. Changing the order is only permitted if one of the factors is real. Let x = xo + i X + j X2 G U be the quaternion associated with the vector x = (xo,Xi,X2)] then the perturbed Kepler problem (3) is given by... 

Jan Vrbik (1995). Perturbed Kepler problem in quaternionic form. J. Phys. A 28, 193-198. 

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. 

Example 1.7 The Kepler problem describes the motion of a body in the plane moving under gravitational force exerted by a second, fixed body (located at the origin) it has the energy E(x,y,x,y) = x ll + y /2--------------7 = - Note that a... 

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