Pre-publicaciones Del Seminario Matematico 2001 Non Linear Stability in Resonant Cases: a Geometrical Approach

Summary. In systems with two degrees of freedom, Arnold's theorem is used for studying nonlinear stability of the origin when the quadratic part of the Hamiltonian is a nondefinite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polynomials in the actions. However, in the case of resonances, it could not be possible to bring the Hamiltonian to the normal form required by Arnold's theorem. In these cases, we determine the stability from analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables.

[1]  G. Bryan,et al.  The Algebra of Invariants The Dynamical Theory of Gases A Treatise on the Analytical Dynamics of Particles and Rigid Bodies , 1905, Nature.

[2]  K. Alfriend.  The stability of the triangular Lagrangian points for commensurability of order two , 1970 .

[3]  A. Elipe,et al.  Oscillators in resonance , 1999 .

[4]  Jesús F. Palacián,et al.  Hamiltonian Oscillators in 1—1—1 Resonance: Normalization and Integrability , 2000, J. Nonlinear Sci..

[5]  Gen-Ichiro Hori,et al.  Theory of general perturbations with unspecified canonical variables , 1966 .

[6]  Jürgen Moser,et al.  Lectures on Celestial Mechanics , 1971 .

[7]  A. Deprit.  The Lissajous transformation I. Basics , 1991 .

[8]  A. Deprit,et al.  STABILITY OF THE TRIANGULAR LAGRANGIAN POINTS , 1967 .

[9]  H. Poincaré,et al.  Les méthodes nouvelles de la mécanique céleste , 1899 .

[10]  Kenneth R. Meyer,et al.  Stability of equilibria and fixed points of conservative systems , 1999 .

[11]  D. Schmidt.  The Stability of the Lagrangian Point L 4 , 1988 .

[12]  A. Markeev.  On the stability of the triangular libration points in the circular bounded three-body problem , 1969 .

[13]  F. G. Gustavson,et al.  Oil constructing formal integrals of a Hamiltonian system near ail equilibrium point , 1966 .

[14]  Jesús F. Palacián,et al.  Reduction of polynomial Hamiltonians by the construction of formal integrals , 2000 .

[15]  K. Meyer,et al.  The Stability of the Lagrange Triangular Point and a Theorem of Arnold , 1986 .

[16]  George Birkhoff.  Physical aspects of dynamical systems , 1927 .

[17]  Victor Szebehely,et al.  Theory of Orbits. , 1967 .

[18]  André Deprit,et al.  Canonical transformations depending on a small parameter , 1969 .

[19]  R. Cushman,et al.  Normal form and representation theory , 1982 .