Lagrange’s mean motion problem
HTML articles powered by AMS MathViewer
- by
S. Yu. Favorov
Translated by: the author - St. Petersburg Math. J. 20 (2009), 319-324
- DOI: https://doi.org/10.1090/S1061-0022-09-01049-8
- Published electronically: February 4, 2009
- PDF | Request permission
Abstract:
The famous mean motion problem, dating back to Lagrange, is about the existence of the average speed for the amplitude of any exponential polynomial with exponents on the imaginary axis, whenever the variable moves along a horizontal line. This problem was completely solved by B. Jessen and H. Tornehave in Acta Mathematica, vol. 77, 1945. Here, we give a simple version of that proof.References
- P. Bohl, Über ein in der Theorie der säkularen Störungen vorkommendes Problem, J. Reine Angew. Math. 135 (1909), 189–283.
- Harald Bohr, Zur Theorie der Fastperiodischen Funktionen, Acta Math. 46 (1925), no. 1-2, 101–214 (German). II. Zusammenhang der fastperiodischen Funktionen mit Funktionen von unendlich vielen Variabeln; gleichmässige Approximation durch trigonometrische Summen. MR 1555201, DOI 10.1007/BF02543859
- F. Bernstein, Über eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem, Math. Ann. 71 (1912), 417–439.
- Raouf Doss, On mean motion, Amer. J. Math. 79 (1957), 389–396. MR 85390, DOI 10.2307/2372687
- Philip Hartman, Mean motions and almost periodic functions, Amer. Math. Soc. 46 (1939), 66–81. MR 0000068, DOI 10.1090/S0002-9947-1939-0000068-9
- Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
- Børge Jessen and Hans Tornehave, Mean motions and zeros of almost periodic functions, Acta Math. 77 (1945), 137–279. MR 15558, DOI 10.1007/BF02392225
- J. L. Lagrange, Théorie des variations séqulaires des éléments des planètes, I, II, Nouveaux Mémoires de l’Académie de Berlin (1781, 1782), Œuvres de Lagrange. T. 5, Gauthier-Villars, Paris, 1870, pp. 123–344.
- B. M. Levitan, Počti-periodičeskie funkcii, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953 (Russian). MR 0060629
- B. V. Shabat, Vvedenie v kompleksnyĭ analiz. Chast′II, 3rd ed., “Nauka”, Moscow, 1985 (Russian). Funktsii neskol′kikh peremennykh. [Functions of several variables]. MR 831938
- Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864
- Hermann Weyl, Mean Motion, Amer. J. Math. 60 (1938), no. 4, 889–896. MR 1507355, DOI 10.2307/2371267
- Hermann Weyl, Mean Motion. II, Amer. J. Math. 61 (1939), no. 1, 143–148. MR 1507367, DOI 10.2307/2371393
Bibliographic Information
- S. Yu. Favorov
- Affiliation: Department of Mathematics, Kharkov National University, Svobody Square 4, Kharkov 61077, Ukraine
- MR Author ID: 189658
- ORCID: 0000-0002-4687-776X
- Email: favorov_s@mail.ru, fav@univer.kharkov.ua
- Received by editor(s): August 10, 2007
- Published electronically: February 4, 2009
- © Copyright 2009 American Mathematical Society
- Journal: St. Petersburg Math. J. 20 (2009), 319-324
- MSC (2000): Primary 33B10, 30B50
- DOI: https://doi.org/10.1090/S1061-0022-09-01049-8
- MathSciNet review: 2424001