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The phase space structure of the extended Sitnikov problem. (English) Zbl 0912.70008

The Sitnikov configuration consists of two primaries of equal mass on Keplerian orbits, while a third mass is confined to oscillate on an axis \((z\)-axis) perpendicular to the primaries’ orbital plane. In this paper the authors treat the extended Sitnikov problem where three bodies of equal masses stay always in the Sitnikov configuration. After some historical remarks, the authors formulate the problem and succeed to reduce the system to a dynamical system with two degrees of freedom having the energy integral \(H\). By means of this reduction, the authors are able to give a qualitative analysis of possible motions in the extended Sitnikov problem and to explore the structure of phace space by using a properly chosen surfaces of section. It is observed that for small energies \(H\) the motion is possible only in small region of phase space, and only thin layers of chaos appear in this region of mostly regular motion. Therefore the authors limit the detailed description of characteristics of chosen surfaces of section to the energy values \(-0.26 \leq H\leq-0.03\). Although the authors compute systematically all surfaces of section between the two energy values mentioned above with a step size \(\Delta H=0.01\), they discuss only those surfaces which show important features. For selected energies the authors also compute the rotation numbers \(\omega\). It is claimed that according to the numerical results it is possible to obtain similar theoretical results, e.g. with the aid of mapping methods.
Reviewer: T.Nono (Hiroshima)

MSC:

70F07 Three-body problems
70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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