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Comptes Rendus Mathématique
Volume 343, n° 7
pages 457-462 (octobre 2006)
Doi : 10.1016/j.crma.2006.09.017
Received : 4 September 2006 ;  accepted : 13 September 2006
Exponential asymptotics and adiabatic invariance of a simple oscillator
Asymptotiques exponentielles et invariance adiabatique dʼun oscillateur simple
 

Chunhua Ou a , Roderick Wong b
a Department of Mathematics and Statistics, Memorial University of Newfoundland, Canada 
b Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong 

Abstract

An alternative proof is provided for Littlewoodʼs asymptotic expression arising from Lorentzʼs problem (1911) on the adiabatic invariance of a simple pendulum. Our approach is based on a standard WKB approximation. Our proof is simpler than those of both Littlewood (1963) and Wasow (1973). If the coefficient function in their differential equation is analytic, then Littlewoodʼs asymptotic expression can even be replaced by an exponentially small term. To cite this article: C.H. Ou, R. Wong, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The full text of this article is available in PDF format.
Résumé

On donne une autre démonstration de lʼexpression asymptotique que Littlewood a obtenue pour le problème de Lorentz (1911) sur lʼinvariance adiabatique dʼun pendule simple. Notre approche repose sur lʼapproximation WKB habituelle. Notre démonstration est plus simple que celle de Littlewood (1963) et celle de Wasow (1973). Si le coefficient de lʼéquation différentielle quʼils considèrent est analytique, alors lʼexpression asymptotique de Littlewood peut même être remplacée par un terme exponentiellement petit. Pour citer cet article : C.H. Ou, R. Wong, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The full text of this article is available in PDF format.


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