Resonance cases and small divisors in a third integral of motion

• DOI: 10.1086/109822
• Publication Date: 2002-07-26T17:51:12Z
• Authors: George Contopoulos; Michael Moutsoulas

This paper contains a complete description of the resonance case A =B, i.e., when unperturbed fre- quencies in two perpendicular directions are equal. The form third integral is different from that non case. secular terms eliminated, step by step, and higher-order calculated means computer. better conserved actual orbits more included it. invariant curves give main characteristics orbits. theoretical in- variant represent sufficiently well empirically found (by orbital cal- culations) up to fourth degree perturbation parameter E included. classification can be achieved even using zero-order integral. For accurate numerical results, however, we need include second or order, especially approaches value for which curve zero velocity opens moving point may go infinity. There three types orbits: A-, B-, C-type Their boundaries numerically some characteristic points series expansions. detailed comparison between theory experiments gives always good agreement sufficient Five periodic have been found, stable unstable. transition also discussed detail. These calculations applied galactic on plane symmetry distorted (nonaxisym- metric) galaxy. If distortion order 20%, find circular become almost rectilinear through central region then reverse sense rotation few billion years. application refers energy exchange coupled oscillators. predicts correct amount exchange. In one oscillators varies 0 2 total energy.