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Une classe d’hamiltoniens polynomiaux isochrones

Published online by Cambridge University Press:  20 November 2018

Bertrand Schuman
Bertrand Schuman*
Affiliation:
Université Pierre etMarie Curie, Paris 6, Tour 46-0, 5ème étage, case 172 4, place Jussieu 75252 Paris Cedex 05 France, courriel: schuman@math.jussieu.fr
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Résumé

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Soit ${{H}_{0}}\,=\,\frac{{{x}^{2}}+{{y}^{2}}}{2}$ un hamiltonien isochrone du plan ${{\mathbb{R}}^{2}}$. On met en évidence une classe d’hamiltoniens isochrones qui sont des perturbations polynomiales de ${{H}_{0}}$. On obtient alors une condition nécessaire d’isochronisme, et un critère de choix pour les hamiltoniens isochrones. On voit ce résultat comme étant une généralisation du caractère isochrone des perturbations hamiltoniennes homogènes considérées dans $\left[ \text{L} \right],\,\left[ \text{P} \right],\,\left[ \text{S} \right]$.

Abstract

Abstract

Let ${{H}_{0}}\,=\,\frac{{{x}^{2}}+{{y}^{2}}}{2}$ be an isochronous Hamiltonian of the plane ${{\mathbb{R}}^{2}}$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian ${{H}_{0}}$ considered in $\left[ \text{L} \right],\,\left[ \text{P} \right],\,\left[ \text{S} \right]$.

Keywords

34C2058F0558F2258F30Hamiltonian systemnormal formsresonancelinearization
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 3 , 01 September 2001 , pp. 323 - 334
Copyright
Copyright © Canadian Mathematical Society 2001

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