About an averaging method which is available in a while resonance zone for non autonomous and non linear classical hamiltonian systems
Description
The aim of this paper is to improve to non-linear classical hamiltonian systems the method which we call ''Resonant Averaging Method'' and which has been previously worked out in linear cases, especially in Quantum Mechanics. We deal with integrable and autonomous system perturbed by a time periodic or pseudo-periodic field and we generalize Bogolioubov and Mitropolsky method. In opposite with other approaches, we are able to give approximated solutions which are available as far from a specific resonance as in its close neighbourhood. It becomes then easy to follow the behaviour of a system by continuity when we vary one of the perturbation frequencies in a large zone which may include a resonance of the non perturbed system. Finally, we illustrate the method in the case of the physical non linear pendulum with a periodic parametric perturbation
Abstract (French)
On etend a des systemes hamiltoniens classiques non-lineaires la methode dite ''des moyennes resonnantes'' anterieurement proposee dans le cas lineaire, notamment en mecanique quantique. Le systeme type cite ici en exemple sera un systeme autonome integrable soumis a une perturbation periodique ou presque periodique par rapport au temps. La methode proposee est une generalisation de la methode des moyennes de Bogolioubov et Mitropolsky et son avantage, par rapport a d'autres methodes, est de fournir une solution approchee qui reste valable aussi bien loin d'une resonance que dans son voisinage immediat. Il devient alors facile de suivre par continuite la reponse du systeme integrable considere (systeme dit ''non perturbe'') lorsqu'on fait varier l'une des frequences partielles de la perturbation dans un large intervalle pouvant inclure une frequence propre du systeme non perturbe. La methode sera illustree par une application a la perturbation parametrique periodique d'un pendule non lineaireAdditional details
Additional titles
- Original title (French)
- Sur une methode de moyennisation valable dans une large zone de resonance pour des systemes hamiltoniens classiques non autonomes et non lineaires
Publishing Information
- Journal Title
- Ann. Fond. Louis de Broglie
- Journal Volume
- 8
- Journal Issue
- 2
- Series
- Ann. Fond. Louis de Broglie.
- Journal Page Range
- 155-186
- ISSN
- 0182-4295
INIS
- Country of Publication
- France
- Country of Input or Organization
- France
- INIS RN
- 15003751
- Subject category
- S71: CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS;
- Descriptors DEI
- CLASSICAL MECHANICS; DISTURBANCES; HAMILTONIAN FUNCTION; JACOBIAN FUNCTION; MATHEMATICAL MODELS; NONLINEAR PROBLEMS; QUANTUM MECHANICS; RESONANCE
- Descriptors DEC
- FUNCTIONS; MECHANICS