A new method to construct integrable approximations to nearly integrable systems in Celestial Mechanics: application to the Sitnikov problem

摘要

The Sitnikov problem for nonzero primaries eccentricities is a non-integrable dynamical system. In this contribution, a second dynamical system close to the original one but being fully integrable is constructed. We denote this system by "approximating integrable system", and we will give a rigorous definition for it as well as for the "distance" between the integrable and the non-integrable system. The first integral of the approximating system is derived in closed form, and from this result, the most important system properties are found algebraically and compared to the ones of the Sitnikov problem obtained by numerical integration. It turns out that for the given range of the eccentricity and initial amplitude, the approximating system describes accurately the most important properties of the Sitnikov problem.