The stability in a strict non-linear sense of a trivial relative equilibrium position in the classical and generalized versions of Sitnikov's problem

Stability, in a strict non-linear sense, of a trivial relative equilibrium position is investigated in the classical and generalized versions of Sitnikov's problem in the case of small eccentricities of the orbits of bodies of finite dimensions. In the classical version (n = 2) of the problem, it is proved that there are no second-, third- and fourth-order resonances and a degenerate case. In the generalized version (2 < n ≤ 5 · 105), it is proved that there are no second- and third-order resonances and a degenerate case. A fourth-order resonance occurs in versions of the problem in which the number of finite size bodies satisfies the inequality 45000 ≤ n ≤ 62597 and the orbital eccentricities e < 0.25. Use of the Arnold–Moser and Markeyev theorems enables one to establish the Lyapunov stability of the trivial positions of relative equilibrium in the above-mentioned versions of Sitnikov's problem.