April  2011, 30(1): 299-312. doi: 10.3934/dcds.2011.30.299

Non-integrability of the collinear three-body problem

1. 

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Received  January 2010 Revised  October 2010 Published  February 2011

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.
Citation: Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299
References:
[1]

R. L. Devaney, Triple collision in the planar isosceles three body problem,, Invent. Math., 60 (1980), 249. doi: 10.1007/BF01390017. Google Scholar

[2]

S. R. Kaplan, Symbolic dynamics of the collinear three-body problem,, in, 246 (1999), 143. Google Scholar

[3]

R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191. doi: 10.1007/BF01390175. Google Scholar

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K. Meyer and Q. Wang, The collinear three-body problem with negative energy,, J. Differential Equations, 119 (1995), 284. doi: 10.1006/jdeq.1995.1092. Google Scholar

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J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113. Google Scholar

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D. G. Saari and Z. Xia, The existence of oscillatory and super hyperbolic motion in newtonian systems,, J. Differential Equations, 82 (1989), 342. doi: 10.1016/0022-0396(89)90137-X. Google Scholar

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M. M. Saito and K. Tanikawa, The rectilinear three-body problem using symbol sequence I: Role of triple collision,, Celest. Mech. Dyn. Astr., 98 (2007), 95. doi: 10.1007/s10569-007-9070-0. Google Scholar

[8]

M. M. Saito and K. Tanikawa, The rectilinear three-body problem using symbol sequence II: Role of the periodic orbits,, Celest. Mech. Dyn. Astr., 103 (2009), 191. doi: 10.1007/s10569-008-9175-0. Google Scholar

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C. Siegel and J. Moser, "Lectures on Celestial Mechanics,", Springer-Verlag, (1971). Google Scholar

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C. Simó, Masses for which triple collision is regularizable,, Celestial Mech., 21 (1980), 25. doi: 10.1007/BF01230243. Google Scholar

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H. Yoshida, A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential,, Physica D, 29 (1987), 128. doi: 10.1016/0167-2789(87)90050-9. Google Scholar

show all references

References:
[1]

R. L. Devaney, Triple collision in the planar isosceles three body problem,, Invent. Math., 60 (1980), 249. doi: 10.1007/BF01390017. Google Scholar

[2]

S. R. Kaplan, Symbolic dynamics of the collinear three-body problem,, in, 246 (1999), 143. Google Scholar

[3]

R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191. doi: 10.1007/BF01390175. Google Scholar

[4]

K. Meyer and Q. Wang, The collinear three-body problem with negative energy,, J. Differential Equations, 119 (1995), 284. doi: 10.1006/jdeq.1995.1092. Google Scholar

[5]

J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113. Google Scholar

[6]

D. G. Saari and Z. Xia, The existence of oscillatory and super hyperbolic motion in newtonian systems,, J. Differential Equations, 82 (1989), 342. doi: 10.1016/0022-0396(89)90137-X. Google Scholar

[7]

M. M. Saito and K. Tanikawa, The rectilinear three-body problem using symbol sequence I: Role of triple collision,, Celest. Mech. Dyn. Astr., 98 (2007), 95. doi: 10.1007/s10569-007-9070-0. Google Scholar

[8]

M. M. Saito and K. Tanikawa, The rectilinear three-body problem using symbol sequence II: Role of the periodic orbits,, Celest. Mech. Dyn. Astr., 103 (2009), 191. doi: 10.1007/s10569-008-9175-0. Google Scholar

[9]

C. Siegel and J. Moser, "Lectures on Celestial Mechanics,", Springer-Verlag, (1971). Google Scholar

[10]

C. Simó, Masses for which triple collision is regularizable,, Celestial Mech., 21 (1980), 25. doi: 10.1007/BF01230243. Google Scholar

[11]

H. Yoshida, A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential,, Physica D, 29 (1987), 128. doi: 10.1016/0167-2789(87)90050-9. Google Scholar

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