Results are obtained about formal stability and instability of Hamiltonian systems with three degrees of freedom, two equal frequencies and the matrix of the linear part is not diagonalizable, in terms of the coefficients of the development in Taylor series of the Hamiltonian of the system. The results are applied to the study of stability of the Lagrangian solutions of the Three Body-Problem in the case in which the center of mass is over the curve ρ * , on the border of the region of linear stability of Routh. The curve ρ * is divided symmetrically in three arcs in such a way that if the center of mass of the three particles lies on the central arc, the Lagrangian solution is unstable in the sense of Liapunov (in finite order), while if the center of mass determines one point that lies on one of the other two arcs of ρ * , then the Lagrangian solution is formally stable.
We present a geometric interpretation of the spectral stability of the triangular libration points in the charged three-body problem. We obtain that the spectral stability varies with the position of the center of mass of the three charges with respect to the circumcenter of the triangle configuration, which does not depend directly of the charges. If the center of mass is outside or on the circumference of a well defined radius ρ, then spectral stability occurs. In addition, we analyze the existence of resonances within the spectral region of stability under this geometric interpretation, determining resonance curves of order 2, 3, 4, . . ., some of them with multiple resonances.
In this work, we consider a Hamiltonian system with three degrees of freedom, whose linear part has all its roots pure imaginary and its three frequencies equal. We determine the kernel of the Lie operator and the normal form, according to Meyer of Hamiltonian in the diagonalizable case and in one of the nondiagonalizable cases, obtaining a normal form of the type obtained by Sokol's kii and Mansilla in previous works.
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