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Méthodes asymptotiques en physique
Titulaire(s) du cours
Gregory KOZYREFF (Coordonnateur) et Fabian BRAUCrédits ECTS
5
Langue(s) d'enseignement
français
Contenu du cours
Theoretical research sometimes resembles gold exploration: the first arrived on a given subject can make every bold simplifying assumption and harvest whatever simple and meaningful result there is to find. By the time the rest of us learn about it, all the good nuggets have been found. Still, some beautiful exact results remain under the surface. Every now and then, they are uncovered through mathematical tours de force. Short of extraordinary mathematical skills, there is, fortunately, a third way towards succesful analytical investigations: Asymptotics – the craft of treating limiting cases. A scientist undertaking an asymptotic study narrows her or his focus on some restricted situations. Mathematically, something must be “small” or something must be “large” and one wishes to make deductions from this observation or hypothesis. This course provides a set of strategies to obtain, when possible, results that are approximately true in some limits. In order to successfully apply asymptotic methods, one first needs to find whether anything at all is “small” or “large”. This can only be done in a systematic way when the problem under study is expressed by dimension- less equations. However, most equations in physics and engineering do not appear in this form and must first be rewritten by expressing the variables within a well-chosen set of units. This preliminary step requires insight and often involves a fair amount of trial and error and we will start the course by a selection of examples. Simple equations that can be solved by elementary techniques will be reviewed. The ultimate goal of asymptotic methods is to reduce the original problem to the study of one of those simpler ones. To this end, we will discuss the methods of matched asymptotic expansion, WKB, and multiple scales, mostly applied to differential equations. If times permits and according to the needs of the students, we will complete the course by asymptotic evaluations of integrals or the asymptotic analysis of discrete problems While this course is mainly for master students in a physics department, it has been conceived with a broader audience in mind: engineers and scientists of all disciplines who at some point want to enrich their numerical investigation by analytical understanding. What is required from the student is to be able to differentiate, integrate, and Taylor-expand elementary functions, and to be familiar with complex analysis. |
Les problèmes mathématiques qui se présentent à un-e chercheur-euse en physique, mathématique ou ingénierie n'ont généralement pas de solution analytique simple. Cependant, il est souvent possible de considérer des cas limites qui, tout en gardant du sens par rapport au point de vue initial, simplifient considérablement l'analyse. Les méthodes asymptotiques présentées dans ce cours ont pour but d'analyser ces cas limites de manière systématique et cohérente. Elle permettent dans le meilleur des cas d'écrire de bonnes approximations analytiques des solutions et dans d'autre cas, de construire des modèles plus simples à étudier numériquement. Une bonne analyse asymptotique est souvent porteuse de sens car elle permet de dégager des tendances et des lois d'échelles et ainsi de se forger une intuition sur des problèmes complexes. De nombreux exemples tirés de la mécanique quantique, de la physique du laser, de la mécanique des fluides ou de l'optique illustreront le cours. |
Objectifs (et/ou acquis d'apprentissages spécifiques)
The aim of this course is to acquire techniques of approximate resolution of research-type differential |
Acquérir les techniques de résolution de problèmes différentiels courants en recherche. |
Pré-requis et Co-requis
Connaissances et compétences pré-requises ou co-requises
It is expected that the student is confortable with differentiation and integration of simple functions. Knowledge of elementary differential equations such as that of the harmonic oscillators or ODEs with constant coefficients is assumed. |
Savoir dériver et intégrer des fonctions simples. Etre capable de résoudre des équations différentielles ordinaire élémentaires telles que l'oscillateur harmonique |
Méthodes d'enseignement et activités d'apprentissages
The course consists of oral lectures alternating with resolutions of exercises, according to the material seen. If necessary the lectures are given in english. |
Cours oral avec alternance de théorie et d'exercices, selon la progression du cours. Si tous les étudiants parlent français, le cours est donné dans cette langues. Sinon, le cours est proposé en anglais. |
Contribution au profil d'enseignement
Références, bibliographie et lectures recommandées
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E. J. Hinch, Perturbation Methods, Cambridge University Press 1991
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C.M. Bender & S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer-Verlag 1999
Support(s) de cours
- Syllabus
Autres renseignements
Contacts
Gregory Kozyreff : 2o6103, ext 5821 (bâtiment NO, 6ème étage, campus plaine)
Campus
Plaine
Evaluation
Méthode(s) d'évaluation
- Examen écrit
- Examen oral
Examen écrit
Examen oral
Langue(s) d'évaluation
- anglais
- français