Comptes Rendus
no. 7
Ordinary Differential Equations/Mathematical Analysis
Exponential asymptotics and adiabatic invariance of a simple oscillator
[Asymptotiques exponentielles et invariance adiabatique d'un oscillateur simple]

Présenté par : Philippe G. Ciarlet

Chunhua Ou1 ; Roderick Wong2
1 Department of Mathematics and Statistics, Memorial University of Newfoundland, Canada
2 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 457-462.

On donne une autre démonstration de l'expression asymptotique que Littlewood a obtenue pour le problème de Lorentz (1911) sur l'invariance adiabatique d'un pendule simple. Notre approche repose sur l'approximation WKB habituelle. Notre démonstration est plus simple que celle de Littlewood (1963) et celle de Wasow (1973). Si le coefficient de l'équation différentielle qu'ils considèrent est analytique, alors l'expression asymptotique de Littlewood peut même être remplacée par un terme exponentiellement petit.

An alternative proof is provided for Littlewood's asymptotic expression arising from Lorentz's 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 Littlewood's asymptotic expression can even be replaced by an exponentially small term.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.09.017

Chunhua Ou 1 ; Roderick Wong 2

1 Department of Mathematics and Statistics, Memorial University of Newfoundland, Canada
2 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong 
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Chunhua Ou; Roderick Wong. Exponential asymptotics and adiabatic invariance of a simple oscillator. Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 457-462. doi : 10.1016/j.crma.2006.09.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.09.017/

[1] J.E. Littlewood Lorentz's pendulum problem, Ann. Phys., Volume 21 (1963), pp. 233-242

[2] R.E. Meyer Adiabatic variation. I. Exponential property for the simple oscillator, Z. Angew. Math. Phys., Volume 24 (1973), pp. 293-303

[3] R.E. Meyer Exponential asymptotics, SIAM Rev., Volume 22 (1980) no. 2, pp. 213-224

[4] F.W.J. Olver Asymptotics and Special Functions, Academic Press, New York, 1974 (Reprinted by, 1997, A.K. Peters, Wellesley, MA)

[5] W. Wasow Adiabatic invariance of a simple oscillator, SIAM J. Math. Anal., Volume 4 (1973), pp. 78-88

[6] R. Wong Asymptotic Approximations of Integrals, Academic Press, Boston, MA, 1989 (Reprinted by, 2001, SIAM, Philadelphia, PA)

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