Abstract
The aim of this paper is to describe the general averaging principle and to discuss the particular case of single-frequency systems, the case of systems with constant frequencies and the case of Hamiltonian systems. We show how the stroboscopic method, which is a method of the nonstandard perturbation theory of differential equations, can be used in this kind of problems. We give various examples which illustrate the simplicity and the effectiveness of the method.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
V.I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Math. 60, (Springer-Verlag, New-York 1989).
V.I. Arnold, V. V. Kozlov, A. I. Neishtadt, Dynamical Systems III, Encyclopaedia of Mathematical Sciences, Vol. 3, (Springer-Verlag, Berlin 1993).
J. L. Callot, T. Sari, Stroboscopie et moyennisation dans les systémes d'équations différentielles à solutions rapidement oscillantes, in : Mathematical Tools and Models for Control, Systems Analysis and Signal Processing,vol. 3, 345 (1983).
S. N. Chow, E. M. de Jager, R. Lutz, The ghost solutions of the logistic equation and a singular perturbation problem, in Advances in Computational Methods for Boundary Layers and Interior Layers, Adv. Numer. Comput. Ser. 6, 15 (1984).
R. Lutz, L'intrusion de 1'analyse non standard dans 1'étude des perturbations singuliéres Asterisque 109-110, 101 (1983).
G. Pascoli, La gravitation,Presses Universitaires de Prance, Coll. Que Sais-je ?, no. 2489, 1989
J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems,Appi. Math. Sciences 58, (Springer-Verlag, New-York 1985).
T. Sari, Sur la théorie asymptotique des oscillations non stationnaires, Astérisque 109-110, 141 (1983).
T. Sari, Systèmes hamiltoniens à paramètres lentement variables, Cahiers Math. Univ. Oran 3, 113 (1987).
T. Sari, Petite histoire de la stroboscopie, in .Colloque Trajectorien a la Mémoire de J. L. Callot et G. Reeb, Strasbourg-Obernai 1995, Publication IRMA, Univ. Strasbourg (1995), 5-15.
T. Sari, Stroboscopy and Averaging, in : Colloque Trajectorien à la Mémoire de J. L. Callot et G. Reeb, Strasbourg-Obernai 1995, Publication IRMA, Univ. Strasbourg (1995), 125-158.
D. R. Smith, Singular-perturbation Theory. An introduction with applications,(Cambridge University Press, Cambridge, 1985).
S. Sternberg, Celestial Mechanics, Part 1, (W. A. Benjamin Inc, New York, 1969).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Sari, T. (2003). Averaging in Hamiltonian Systems with Slowly Varying Parameters. In: Macias, A., Uribe, F., Diaz, E. (eds) Developments in Mathematical and Experimental Physics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0199-2_9
Download citation
DOI: https://doi.org/10.1007/978-1-4615-0199-2_9
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-4963-1
Online ISBN: 978-1-4615-0199-2
eBook Packages: Springer Book Archive