From the Birkhoff-Gustavson normalization to the Bertrand-Darboux integrability condition
Yoshio Uwano
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- cubic polynomial potential and with a homogeneous- 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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