Isoenergetic families of quasi-periodic solutions near the equilateral solution of the three-body problem

Abstract

The tensor\(C_k^i = R_{jkl}^i \dot q^j \dot q^l \) appearing in the equation of geodesic deviation is computed for the equilateral solution of the general three-body problem. The eigenvalues and eigenvectors ofCki are determined. It is found that at least one of the eigenvalues is negative, irrespective of the masses of the bodies. This implies that the equilateral solution is not stable. The eigenvectors with positive eigenvalues generate isoenergetic 1-parameter families of quasi-periodic solutions in the neighborhood of the equilateral solution. The relation between the 1-parameter families constructed here and those known from the literature is discussed.