From the Birkhoff-Gustavson normalization to the Bertrand-Darboux integrability condition
arXiv:nlin/0005027 · doi:10.1088/0305-4470/33/37/315
Abstract
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.
23 pages, LaTeX (iop.sty), typos and Appendix added