Modification of Bertrand’s Theorem and Extended Runge-Lenz Vector
WU Zuo-bing1,2, ZENG Jin-yan1
1Department of Physics, Peking University, Beijing 100871
2Laboratory for Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080
Modification of Bertrand’s Theorem and Extended Runge-Lenz Vector
WU Zuo-bing1,2;ZENG Jin-yan1
1Department of Physics, Peking University, Beijing 100871
2Laboratory for Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080
Abstract： It is shown that for a particle with suitable angular momenta 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 O3. The closeness of a planar orbit implies the radial and angular motional frequencies are commensurable.
1999, Vol. 16 
