# On the Metric Stability and the Nekhoroshev Estimate of the Velocity of Arnold Diffusion in a Special Case of the Three-body Problem

*2021, Volume 26, Number 4, pp. 321-330*

Author(s):

**Markeev A. P.**

A study is made of the stability of triangular libration points in the nearly-circular
restricted three-body problem in the spatial case. The problem of stability for most (in the sense
of Lebesgue measure) initial conditions in the planar case has been investigated earlier. In the
spatial case, an identical resonance takes place: for all values of the parameters of the problem
the period of Keplerian motion of the two main attracting bodies is equal to the period of
small linear oscillations of the third body of negligible mass along the axis perpendicular to the
plane of the orbit of the main bodies. In this paper it is assumed that there are no resonances
of the planar problem through order six. Using classical perturbation theory, KAM theory
and algorithms of computer calculations, stability is proved for most initial conditions and the
Nekhoroshev estimate of the time of stability is given for trajectories starting in an addition to
the above-mentioned set of most initial conditions.

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