Planar oscillations of a dumb-bell of a variable length in a central field of Newtonian attraction. Exact approach

•Constant attitude configurations are possible for the dumbbell of variable length in an elliptic Kepler-like orbit.•The problem under investigation makes it possible to proceed with exact dynamic formulation.•Results on stability are possible to be obtained analytically using symmetric formulae.

Dynamics of a dumb-bell of a variable length in a central field of Newtonian attraction is considered. It is assumed that the body moves in a plane fixed in the absolute space and passing through an attracting center. The law of length׳s variation providing an existence of stationary configurations is pointed out. For these configurations the dumb-bell forms a constant angle with a local vertical passing through the center of mass of the dumb-bell, which moves in an elliptic orbit similar to the Keplerian. In particular, the mentioned constant angle may be equal to zero. In contrast to previous investigations (Burov and Kosenko, 2011 [8] and [10], Burov, 2011 [9]) the problem is solved within the exact formulation, without supplementary simplifying assumptions concerning smallness of the dumb-bell in comparison to its distance from the attracting center.