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Modular-function representation of the solutions of the problem of two fixed centres. (English. Russian original) Zbl 0808.70003
J. Appl. Math. Mech. 57, No. 6, 953-963 (1993); translation from Prikl. Mat. Mekh. 57, No. 6, 3-13 (1993).
Summary: The problem of two fixed centres can be integrated in quadratures by using Stoeckel’s theorem in spheroidal coordinates. The elliptic coordinates of the meridian plane satisfy equations that define a complex elliptic curve. The solution can be written down by using the uniformization formula of the elliptic curve. The constants defining the result in terms of doubly periodic functions are calculated successively from the initial data. The half-period ratio is determined by solving a complex transcendental equation using modular functions. The magnitude of the half-period of least absolute value is determined using \(\vartheta\)- functions of the zero argument and using invariants of the Weierstrass \(\wp\)-function.
MSC:
70F99 Dynamics of a system of particles, including celestial mechanics
33E05 Elliptic functions and integrals
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References:
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