Abstract
Using a method previously applied to the treatment of the Mathieu differential equation, we solve the Hill's differential equation of lunar theory through the way of operational calculus, which avoids the cumbersome infinite determinants of the classical procedure. The one-sided Laplace transformation changes it into a difference equation with an infinite number of terms and variable coefficients. When its first member is divided by a suitable factor, this difference equation is the image of an integral equation of the Volterra type which is equivalent to the initial Hill's differential equation. Solution of this Volterra integral equation is unique and it is the general solution of the Hill's differential equation. This solution is a series in the powers of a small dimensionless parameter∈ 2 which appears as a factor in the second member of the Hill's differential equation. We reduce it to the sum of its terms of degree ≤12 with respect to ɛ which is the precision usually required in a lunar theory and we write down effectively the coefficients of the terms in∈ 2, (∈ 2)2 and the coefficient of the term in (∈ 2)3 which depends upon the initial valuey(0) of the Hill's differential equation.
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Brouwer, D. and Clemence, G. M.: 1961,Methods of Celestial Mechanics, Chapter 12, pp. 335–373, Academic Press Inc., New York.
Giacaglia, G. E. O.: 1972,Perturbation Methods in Non-Linear Systems, Vol. 8, Chapter 3, pp. 201–203, Applied Mathematical Science, Springer Verlag, New York, Heidelberg, Berlin.
McLachlan, N. W. and Humbert, P.: 1941,Formulaire pour le calcul symbolique, Memorial des Sciences Mathematiques, fascicule C, Gauthier Villars, Paris.
Parodi, M.: 1957,Introduction a l'étude de l'analyse symbolique, Gauthier Villars, Paris.
Parodi, M.: 1958, ‘Equations intégrales et équations du type de Mathieu’,Journal de Mathématiques Pures et Appliquées, serie 9, tome 37, pp. 45–54, Gauthier Villars, Paris.
Poincaré, H.: 1900,Bulletin Astronomique, tome 17, pp. 134–143, Gauthier Villars, Paris.
Stumpff, K., with the collaboration of Meffroy, J.: 1974,Himmelsmechanik, Vol. 3, Chapter 34, pp. 456–497, Veb, Deutscher Verlag der Wissenschaften, Berlin.
Tisserand, F.: 1894,Traité de Mécanique Céleste, Vol. 3, Chapter 2, pp. 18–26, Gauthier Villars, Paris.
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Meffroy, J. On the solving of the Hill's differential equation through the method of operational calculus. The Moon and the Planets 23, 73–97 (1980). https://doi.org/10.1007/BF00897581
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DOI: https://doi.org/10.1007/BF00897581