Abstract
The main theorems of the theory of averaging are formulated for slowly varying standard systems and we show that it is possible to extend the class of perturbation problems where averaging might be used.
The application of the averaging method to the perturbed two-body problem is possible but involves many technical difficulties which in the case of the two-body problem with variable mass are avoided by deriving new and more suitable equations for these perturbation problems.
Application of the averaging method to these perturbation problems yields asymptotic approximations which are valid on a long time-scale. It is shown by comparison with results obtained earlier that in the case of the two-body problem with slow decrease of mass the averaging method cannot be applied if the initial conditions are nearly parabolic.
In studying the two-body problem with quick decrease of mass it is shown that the new formulation of the perturbation problem can be used to obtain matched asymptotic approximations.
Резюме
Основные теоремы теории осреднения формулируются для меденно изменяющихся стандаптных систем и показано, что возможно расширение класса проблем теории возмущений, к которым применим метод осреднения.
Приложение метода осреднения к возмущённой задаче двух тел является возможным, однако, оно связано с многими техническими трудностями, которые в случае задачи двух тел с переменными массами устраняются выводом новых уравнений, более пригодных для этих задач.
При применении метода осреднения к указанным задачам получаются асимптотичэские приближения, пригодные на длительных интервалах времени.
Сравнение с резуьтатами, полученными ранее, показывает, что в случае задачи двух тел с медленно уменьшающимися массами метод осреднения неьзя применять, если начальные условия соответствуют почти параболическому движению. При изучении задачи двух тэл с быстро уменшающимися массами показано, что новая формулировка задачи о возмущениях может быть исползована для получения согласованных асимптотических приближений.
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Verhulst, F. Asymptotic expansions in the perturbed two-body problem with application to systems with variable mass. Celestial Mechanics 11, 95–129 (1975). https://doi.org/10.1007/BF01228739
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DOI: https://doi.org/10.1007/BF01228739