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The problem of the spherical pendulum from the standpoint of periodic solutions

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Rendiconti del Circolo Matematico di Palermo (1884-1940)

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  1. Read before the Chicago Section of the American Mathematical Society, April 8, 1910.

  2. Research Associate of the Carnegie Institution of Washington.

  3. G. Kobb Sur le mouvement ďun point matériel sur une surface de révolution [[cta Mathematica, Vol. X (1887), pp. 89–108] has proved that there are only five surfaces (including the sphere) for which the solution of the problem can be expressed in terms of elliptic integrals.O. Staude Über die Bewegung eines schweren Punetes auf einer Rotationsfläche [Acta Mathematica, Vol. XI (1888), pp. 303-332] has discussed the properties of motion on more general surfaces, limiting his investigation, however, to surfaces of revolution with vertical axes which are intersected twice, at most, by ahorizontal plane.

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  4. An altogether different treatment of the problem of the spherical pendulum, using theJacobi elliptic functions, is given byA. Tissot Thèsede Mécanique [Journal de Mathématiques pures et appliquées, Series I, Vol. XVII (1852), pp. 88–116] and the results will be found reproduced in many works on elliptic functions.Halphen [Traité des fonctions elliptiques et de leurs applications (Paris, Gauthier-Villars), Vol. II (1888), p. 126], gives a treatment in the Weierstrass elliptic functions.

  5. H. Poincaré,Les Méthodes nouvelles de la Mécanique céleste (Paris, Gauthier-Villars), Vol. I (1892), p. 58.

  6. See Poincaré, loc. cit. 5), Vol. I, Chap. III.

  7. Since the solution is periodic it can be represented as a Fourier series. The coefficients are defined by integrals which can not be easily obtained. See Weierstrass,Ueber eine Gattung reell periodischer Functional [Monatsberichte der Kgl. Preussischen Akademie der Wissenschaiten zu Berlin, Jahrgang 1866, pp. 97–115, 185]. — The construction of the solution of (12) for 8 = 0 as a power series in x2 was given by M. Désiré André,Sur le développement de la fonction elliptique X(x) suivant les puissances croissantes du module [Annales scientifiques de ľEcole Normale supérieure (Paris), Series It, Vol. VIII (1879), PP– 151–168]. Since & was not introduced the periodic character of the solution was not in evidence and the range of its validity was not determined.

  8. The coefficient being an elliptic function is in fact doubly periodic. This equation was treated with reference to this double periodicity by Hermite,Sur quelques applications des fonctions elliptiques (Paris, Gauthier-Villars, 1885), particularly on pp. 109 and seq.

  9. G. W. Hill,On the part of the motion of the lunar perigee which is a function of the mean motions of the sun ani the moon [Acta Mathematica, Vol. VIII (1886), pp. 1–36.—The Collected Mathematical Works (Published by the Carnegie Institution of Washington, 1905; with an Introduction byH. Poincaré), Vol. I, pp. 243–270.

  10. F. Lindemann,Deber die Differentialgleichung der Functionen des elliptischen Cylinders [[athematische Annalen, Vol. XXII (1883), pp. 117–123].

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  11. A. Lindstedt,Ueber die allgemeine Form der Integrale des Dr exkörperproblems [Astronomische Nachrichten, No. 2503 (1883)].

  12. H. Bruns,Ueber eine Differentialgleichung der Störungstheorie [Astronomische Nachrichten, Nos. 2533 (1883), 2553 (1884)].

  13. O. Callandreau,Sur une équation différentielle de la théorie des perturbations et remarques relatives aux Nos. 2389 et 2435 des Astr. Nachr. [Astronomische Nachrichten, No. 2547 (1884)].

  14. T. J. Stieltjes,Quelques remarques sur ľintégration ďune équation différentielle [Astronomische Nachrichten, Nos. 2602 (1884)].

  15. P. Harzer,Ueber eine Differentialgleichung der Störungstheorie [Astronomische Nachrichten, Nos. 2850, 2851 (1888)].

  16. F. R. Moulton andW. D. Macmillan,On the Solutions of Certain Types of Linear Differential Equations with Periodic Coefficients [[merican Journal of Mathematics, Vol. XXXIII (1911), pp. 63–96].

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  17. F. Tisserand Recherches concernant ľéquation différentielle \(\frac{{d^2 x}}{{d t^2 }} + x\left( {q^2 + 2q_1 \cos 2t} \right) = 0\) [[Bulletin Astronomique, Vol. IX (1892), pp. 102–112] has treated this case for a special equation in great detail.

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  18. G. W. Hill, loc. cit. 9). — See alsoF. Tisserand,Traité de Mécanique céléste ([aris, Gauthier-Villars), Vol. III (1894), Chap. I.

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  19. G. H. Darwin,Periodic Orbits [[cta Mathematica, Vol. XXI (1897), pp. 99–242].

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  20. É. Picard,Traité ďAnalyse (Paris, Gauthier-Villars), Vol. II (2e édition, 1905), Chap. XI.

  21. Various means of approximating the general solution have been devised. See:H. Resal,Traité de Mécanique générale ([aris, Gauthier-Villars), Vol. I (1895), p. 180;F. Tisserand,Sur le mouvement du pendule conique [Bulletin des Sciences Mathématiques et Astronomiques, Series II, Vol. V (1881), pp. 448–454];de Sparre,Sur le développement en série des formules du mouvement du pendule conique et sur quelques propriétés de ce mouvement [Annales de la Société scientifique de Bruxelles, Vol. XVI B (1892), pp. 181-202].

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  22. See paper byMoulton andMacMillan, loc. cit.16), p. 78.

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    F. R. Moulton

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Read before the Chicago Section of the American Mathematical Society, April 8, 1910.

Research Associate of the Carnegie Institution of Washington.

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Moulton, F.R. The problem of the spherical pendulum from the standpoint of periodic solutions. Rend. Circ. Matem. Palermo 32, 338–364 (1911). https://doi.org/10.1007/BF03014805

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