From the Birkhoff-Gustavson normalization to the Bertrand-Darboux integrability condition

Details From the Birkhoff-Gustavson normalization to the Bertrand-Darboux integrability condition From the Birkhoff-Gustavson normalization to the Bertrand-Darboux integrability condition

2000-05-15

arxiv.org/abs/nlin/0005027v3

Journal of Physics A, vol.33, 6635-6653 (2000)

Nonlinear Sciences

Exactly Solvable and Integrable Systems

23 pages, LaTeX (iop.sty), typos and Appendix added

Scientific paper

10.1088/0305-4470/33/37/315

The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the Birkhoff-Gustavson(BG)-normalization: By solving an inverse problem of the BG-normalization on computer algebra, it is shown that if the perturbed harmonic oscillators with a homogeneous-{\it cubic} polynomial potential and with a homogeneous-{\it quartic} polynomial potentials admit the same BG-normalization up to degree-4 then both oscillators satisfy the Bertrand-Darboux integrability condition.

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