Kepler problem, perturbed

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 the preceding text we have presented a unified theory of regularization of the perturbed Kepler motion. Quaternion algebra allows for an elegant treatment of the spatial case in a way completely analogous to the way the planar case is traditionally handled by means of complex numbers. As a consequence of the linearity of the regularized equations of the perturbed Kepler motion, the problem of satellite encounters reduces to a linear perturbation problem, the problem of coupled harmonic oscillators. Orbital elements based on the oscillators may lead to a simpified discussion of ordered and chaotic behavior in repeated satellite encounters. This has been demonstrated by means of an instructive example. 

The development in celestial mechanics after Newton was largely in the hands of the French mathematician Pierre-Simon Laplace (1749-1827). The stability of the solar system was the major unsolved problem. Neither Kepler s laws nor Newton s mechanics could be applied successfully to more than a single orbit at a time. The imiversal law of gravitation must clearly apply to any pair of celestial bodies and with several planets and moons circling the sun it is inevitable that mutual perturbations of the predicted perfect elliptical orbits should occur. Newton himself could never precisely model not even the lunar motion and concluded that divine intervention was periodically necessary to maintain the equilibrium of the solar system. 

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