Abstract
Some connections between the methods indicated in the title are considered. Thus, for example, it is shown that the solvability in periodic functions of the equations of the zeroth approximation of the WKB method is equivalent (in the completely integrable case) to the vanishing of the variation with respect to the variables of the action of the averaged Lagrangian.
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Literature cited
S. Yu. Dobrokhotov and V. P. Maslov, “Finite-zone almost-periodic solutions in the WKB approximations,” in: Sovr. Probl. Mat.,15 (Itogi Nauki i Tekhniki, VINITI AN SSSR), Moscow (1980), pp. 3–94.
V. I. Arnol'd, Mathematical Methods of Classical Mechanics [in Russian], Moscow (1974).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 162–171, 1981.
In conclusion, the author would like to express his sincere thanks to V. M. Babich for his attention to this work and for useful discussions and, although this is not honored by tradition, to S. Yu. Dobrokhotov and V. P. Maslov whose work “Finite-Zone Almost Periodic Solutions in the WKB Approximations” served as the foundation and stimulus for writing the present note.
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Kurylev, Y.V. A nonlinear analog of the WKB method and the method of the averaged Lagrangian. J Math Sci 24, 367–372 (1984). https://doi.org/10.1007/BF01086996
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DOI: https://doi.org/10.1007/BF01086996