Separation of variables from the Birkhoff-Gustavson normalization viewpoint Yoshio Uwano (uwano@amp.i.kyoto-u.ac.jp)
Abstract The Bertrand-Darboux theorem (BDT) has been well-known as a key to the integrability and the separability of simple dynamical systems on the Euclidean plane. In the paper [1], a new deep relation was found between the conditions of the BDT for the perturbed harmonic oscillators (PHOs) with qubic homogeneous polynomial potentials and for the PHOs with quartic ones, which is an outcome of \lq the inverse problem' of the Birkhoff-Gustavson (BG) normalization of those PHOs. The aim of this talk is report that the relation found in [1] can be extended to the cases between the PHOs with homogeneous polynomial potentials of degree $r$ ($r$-PHOs) and the PHOs with the homogeneous plynomial potentials of degree-$2(r-1)$ ($2(r-1)$-PHOs) for any odd $r$ ($\geq 3$): \par\smallskip\noindent {\it An $r$-PHO share the same BG normal form up to degree-$2(r-1)$ with a $2(r-1)$-PHO if and only if the $r$-PHO is separable within a rotation of Cartesian coordinates. Further, the $2(r-1)$-PHO is separable in the same rotation of Cartesian coordinates.} \par\smallskip\noindent The separability of the $r$-PHOs is hence characterized from the BG normalization viewpoint. \par\bigskip\noindent [1] Y.Uwano, J.~Phys.~{\bf A33}, 6635-53 (2000). |