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Non-integrability of the collinear three-body problem

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  • We consider the collinear three-body problem where the particles 1, 2, 3 with masses $m_{1}, m_{2}, m_{3}$ are on a common line in this order. We restrict the mass parameters to those that satisfy an "allowable" condition. We associate the binary collision of particle 1 and 2 with symbol "1", one of 2 and 3 with "3", and the triple collision with "2", and then represent the patterns of collisions of orbits in the time evolution using the symbol sequences. Inducing a tiling on an appropriate cross section, we prove that for any symbol sequence with some condition, there exists an orbit realizing the sequence. Furthermore we show the existence of infinitely many periodic solutions. Our novel result is the non-integrability of the collinear three-body problem with the allowable masses.
    Mathematics Subject Classification: Primary: 70F07, 37J30 ; Secondary: 70F16, 37B10.


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