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On the solvability of an equation describing the periodic motions of a satellite in its elliptic orbit. (English) Zbl 0581.70024
Yu. D. Shlapak hat bewiesen, daß das Randwertproblem \(x''+f(t,x,x',x'')=y(t)\), \(x^ j(0)=x^ j(2\pi)\), \(j=0,1\) eine periodische Lösung besitzt, wenn f(t,p,q,r) Lipschitzbedingungen in p,q und r genügt [Ukr. Mat. Zh. 26, 850-854 (1974; Zbl 0314.34049)]. Der erste Autor hat diese Aussage verschärft [Nonlinear Anal. 4, 259-281 (1980; Zbl 0444.47046) und die Autoren, J. Math. Anal. Appl. 89, 462-488 (1982; Zbl 0516.34019)]. Die Ergebnisse werden hier auf den speziellen Fall \(f=\alpha \sin x-2e \sin tx'+e \cos tx''\) und \(y=4e \sin t\) angewendet. Die Resultate führen zu Stabilitätsaussagen über die periodische Bewegung eines Satelliten auf elliptischer Bahn und zu einem besseren Verständnis des Stabilitätsbereiches, ausgedrückt in den Veränderlichen e, der Exzentrizität der elliptischen Bahn, und \(\alpha\), dem Impuls des Satelliten.
Reviewer: F.Selig

MSC:
70M20 Orbital mechanics
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References:
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