[Asymptotiques exponentielles et invariance adiabatique d'un oscillateur simple]
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.
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@article{CRMATH_2006__343_7_457_0,
author = {Ou, Chunhua and Wong, Roderick},
title = {Exponential asymptotics and adiabatic invariance of a simple oscillator},
journal = {Comptes Rendus. Math\'ematique},
pages = {457--462},
publisher = {Elsevier},
volume = {343},
number = {7},
year = {2006},
doi = {10.1016/j.crma.2006.09.017},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2006.09.017/}
}
TY - JOUR AU - Ou, Chunhua AU - Wong, Roderick TI - Exponential asymptotics and adiabatic invariance of a simple oscillator JO - Comptes Rendus. Mathématique PY - 2006 SP - 457 EP - 462 VL - 343 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2006.09.017/ DO - 10.1016/j.crma.2006.09.017 LA - en ID - CRMATH_2006__343_7_457_0 ER -
%0 Journal Article %A Ou, Chunhua %A Wong, Roderick %T Exponential asymptotics and adiabatic invariance of a simple oscillator %J Comptes Rendus. Mathématique %D 2006 %P 457-462 %V 343 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2006.09.017/ %R 10.1016/j.crma.2006.09.017 %G en %F CRMATH_2006__343_7_457_0
Ou, Chunhua; Wong, Roderick. Exponential asymptotics and adiabatic invariance of a simple oscillator. Comptes Rendus. Mathématique, Tome 343 (2006) no. 7, pp. 457-462. doi : 10.1016/j.crma.2006.09.017. http://www.numdam.org/articles/10.1016/j.crma.2006.09.017/
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