La Forma Normal De Lie en Un Caso Críitico De Un Sistema Hamiltoniano Con Tres Grados De Libertad Y Tres Frecuencias Iguales

Abstract

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.

Stability of Hamiltonian Systems with Three Degrees of Freedom and the Three Body-Problem

“…The demonstrations are adaptation of what the author did in Mansilla (2004), in a case of a Hamiltonian system with three degrees of freedom and three nonnull equal frequencies and nondiagonalizable linear part.…”

“…We find the symplectic transformation, dependent of µ, which takes the quadratic part of the Hamiltonian to normal form, determine the coefficient h 020 and prove Theorem 1.2. In the Appendix, we present the results used in the demonstration of the Theorem 1.2 (whose demonstrations are similar to those obtained in Mansilla, 2004), refered to the normal Lie form of the Hamiltonian system with three degrees of freedom, two of which with equal non null frequencies and nondiagonalizable linearized system and we write the coefficients of the normalization matrix.…”