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    On the Dumb-Bell Equilibria in the Generalized Sitnikov Problem

    Received 24 October 2022; accepted 21 November 2022; published 14 December 2022

    2022, Vol. 18, no. 4, pp.  577-588

    Author(s): Krasil'nikov P. S., Ismagilov A. R.

    This paper discusses and analyzes the dumb–bell equilibria in a generalized Sitnikov problem. This has been done by assuming that the dumb–bell is oriented along the normal to the plane of motion of two primaries. Assuming the orbits of primaries to be circles, we apply bifurcation theory to investigate the set of equilibria for both symmetrical and asymmetrical dumb–bells.
    We also investigate the linear stability of the trivial equilibrium of a symmetrical dumb–bell in the elliptic Sitnikov problem. In the case of the dumb–bell length $l\geqslant 0.983819$, an instability of the trivial equilibria for eccentricity $e \in (0,\,1)$ is proved.
    Keywords: Sitnikov problem, dumb–bell, equilibrium, linear stability
    Citation: Krasil'nikov P. S., Ismagilov A. R., On the Dumb-Bell Equilibria in the Generalized Sitnikov Problem, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp.  577-588

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