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

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

Apr 1981

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1981cemec..23..383v&link_type=abstract

Celestial Mechanics, vol. 23, Apr. 1981, p. 383-395.

Other

Eigenvalues, Eigenvectors, Geodesy, Isoenergetic Processes, Periodic Variations, Three Body Problem, Celestial Mechanics, Equations Of Motion, Hypergeometric Functions, Linearization, Surface Geometry

Scientific paper

Isoenergetic families of solutions to the three-body problem close to the equilateral solution are analyzed with the use of the equation of geodesic deviation. The equation of geodesic deviation for a conservative system is introduced, and it is shown that each positive eigenvalue of the tensor appearing in the equation defines a one-parameter family of geodesics that oscillate around the original geodesic, all of which lie on the same energy hypersurface. Calculations of the eigenvalues and eigenvectors of the tensor around the equilateral solution of the three-body problem is shown to result in at least one negative eigenvalue, which implies that the equilateral solution is not stable. The eigenvectors with positive eigenvectors, on the other hand, generate isoenergetic one-parameter families of quasi-periodic solutions in the neighborhood of the equilateral solution. Comparison of the one-parameter families obtained with the equation of geodesic deviation with those obtained by linearization of the equations of motion in the rotating coordinate frame reveals the linearization procedure in general not to yield isoenergetic families.

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