Modification of Bertrand's theorem and extended Runge-Lenz vector

It is shown that for a particle with suitable angular moments in the screened Coulomb potential or isotropic harmonic potential, there still exist closed orbits rather than ellipse, characterized by the conserved aphelion and perihelion vectors, i.e. extended Runge-Lenz vector, which implies a higher dynamical symmetry than the geometrical symmetry O-3. The closeness of a planar orbit implies the radial and angular motional frequencies are commensurable.

经典轨道的封闭性和径向Schrodinger方程的因式分解

研究表明，保证经典轨道具有封闭性的Bertrand定理可以进一步推广，在适当的角动量睛，仍存在着非椭圆的闭合轨道。对于屏蔽Coulomb场，可获得广义Runge-Lenz矢量。这种轨道封闭性与径向Schrodinger方程因式分解相对应。