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By Yusuke Hagihara

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Extra resources for Celestial mechanics. Vol. 5, Part 2. Topology of the three-body problem.

Sample text

1 be a continuous transformation which carries a variable point P of S into a variable point P of the same surface. 1 is of the so-called s-to-one type. 1 from S to S is everywhere oneto-one. Then by Euler's formula (Section 24. 7) IXo - IX1 + IX2 = 3 - K we compare the connectivity tc* of the covering surface S* with the connectivity ;c of S and obtain 3 - tc* = s(3 - ;c) - b, ( 6) where b is the number of equivalent simple branch points of the covering surface. But since S* and S are each images of S we have tc* = ;c = tc, and (6) reduces to (s - 1)(3 - tc) - b = 0.

By a selected determination of the transformation T in this plane all other transformations are obtained by a translation in the 8-direction through any distance 2krr (k = 1, 2, ... ). The circle C appears as a straight line r = a, parallel to the {}-axis. r and rl appear as open curves lying above this line and extending indefinitely far to the right and left, while C 1 , C 2 , • • • are similar curves, C 1 lying above C, C 2 above C 1, and so on. All these curves are congruent in each interval 2krr ~ {) ~ 2(k + l)ir.

The original canonical system (K) is represented inside the circle K by a system of arcs terminating either inside K and at the point C or on the circumference K itself. Two or more of these arcs are always needed to represent an entire curve of the system (K). The situation is illustrated by Figure 12, which represents the case of a relatively simple transformation of a surface of genus 2. ::. i such that Ri is not cut by any of the curves (K) on S and is hence represented in the plane by a single connected piece.

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