Anisotropic Kepler and anisotropic two fixed centres problems     

Andrzej J. Maciejewski  Maria Przybylska  Wojciech Szumiński 《Celestial Mechanics and Dynamical Astronomy》2017,127(2):163-184

In this paper we show that the anisotropic Kepler problem is dynamically equivalent to a system of two point masses which move in perpendicular lines (or planes) and interact according to Newton’s law of universal gravitation. Moreover, we prove that generalised version of anisotropic Kepler problem as well as anisotropic two centres problem are non-integrable. This was achieved thanks to investigation of differential Galois groups of variational equations along certain particular solutions. Properties of these groups yield very strong necessary integrability conditions.  相似文献