Non-Integrability of a weakly integrable Hamiltonian system

Details Non-Integrability of a weakly integrable Hamiltonian system Non-Integrability of a weakly integrable Hamiltonian system

2003-08-29

arxiv.org/abs/nlin/0308033v1

Nonlinear Sciences

Chaotic Dynamics

24 pages, accepted for publication on Celestial Mechanics and Dynamical Astronomy

Scientific paper

10.1023/B:CELE.0000016820.95989.

The geometric approach to mechanics based on the Jacobi metric allows to easily construct natural mechanical systems which are integrable (actually separable) at a fixed value of the energy. The aim of the present paper is to investigate the dynamics of a simple prototype system outside the zero-energy hypersurface. We find that the general situation is that in which integrability is not preserved at arbitrary values of the energy. The structure of the Hamiltonian in the separating coordinates at zero energy allows a perturbation treatment of this system at energies slightly different from zero, by which we obtain an analytical proof of non-integrability.

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