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Celestial mechanics. Vol. 5, Part. 1. Topology of the by Yusuke Hagihara

By Yusuke Hagihara

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Additional resources for Celestial mechanics. Vol. 5, Part. 1. Topology of the three-body problem.

Example text

Thus we find, taking one of the two branches, w= -18v·~1(l + \131), EXISTENCE OF SOLUTIONS where ~l denotes a series in powers of eu 'Y)1, vanishing for e1 = 'YJ1 = 0. The coefficients of W and also W itself are periodic functions of a. J&; [g cos a + 'YJ sin a][l + ~(g, r;, a)], (21) e where ~ is periodic in a and vanishes for = 'Yj = 0 and is regular in a domain D of g, 1J independently of a. Let the velocity of P relative to S be v such that 1 2 {II;+2r+µ 1 2 v} 2v- = -C, or and put {a__w + -1=-I aw _ 2-1=1 p (g + -1=-I YJ)}/c2pv) 2 ae i11J =exp {-l=t a}·(l + 0), where O(g, YJ, a) is a power series in g, 'YJ of the same nature not necessarily real, similarly to ~- Consider + -1=-I aw {aw ae ar; 2-l=t pz(g + -1=-I r;)}/c2pv) as~, = k, but (22) e, where the modulus of k is unity, and 7J are regarded as parameters with their initial values g0 , YJo, but a is unknown.

And e remain finite as r - 0, and that

2:ir). [}.. [} -=----p--. [}0 , regular for all values of u for p = 0. [}0 = constant, u = 0, so that their right-hand members vanish for u = 0. [}0 and regular for p = 0 for any finite value of u. Thus we can prove the theorem that the system (8) does not admit any solution other than the hok>morphic integral that satisfies lim s(p) = -1. (13) p-0 A holomorphic integral of (8) corresponds to an oo 1 solutions of (8b) for which u(p) identically vanishes, or lim u(p) = 0. (13a) p-0 Suppose that there were another solution of (8b) for which u(p) would satisfy (13a).

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