# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 249-267. doi: 10.1007/BF01390017. [2] S. R. Kaplan, Symbolic dynamics of the collinear three-body problem, in "Geometry and Topology in Dynamics" (eds. M. Barge and K. Kuperberg), Contemp. Math., 246, Amer. Math. Soc., (1999), 143-162. [3] R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227. doi: 10.1007/BF01390175. [4] K. Meyer and Q. Wang, The collinear three-body problem with negative energy, J. Differential Equations, 119 (1995), 284-309. doi: 10.1006/jdeq.1995.1092. [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), 113-120. [6] D. G. Saari and Z. Xia, The existence of oscillatory and super hyperbolic motion in newtonian systems, J. Differential Equations, 82 (1989), 342-355. doi: 10.1016/0022-0396(89)90137-X. [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), 95-120. doi: 10.1007/s10569-007-9070-0. [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), 191-207. doi: 10.1007/s10569-008-9175-0. [9] C. Siegel and J. Moser, "Lectures on Celestial Mechanics," Springer-Verlag, New York-Heidelberg, 1971. [10] C. Simó, Masses for which triple collision is regularizable, Celestial Mech., 21 (1980), 25-36. doi: 10.1007/BF01230243. [11] H. Yoshida, A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica D, 29 (1987), 128-142. doi: 10.1016/0167-2789(87)90050-9.

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##### References:
 [1] R. L. Devaney, Triple collision in the planar isosceles three body problem, Invent. Math., 60 (1980), 249-267. doi: 10.1007/BF01390017. [2] S. R. Kaplan, Symbolic dynamics of the collinear three-body problem, in "Geometry and Topology in Dynamics" (eds. M. Barge and K. Kuperberg), Contemp. Math., 246, Amer. Math. Soc., (1999), 143-162. [3] R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227. doi: 10.1007/BF01390175. [4] K. Meyer and Q. Wang, The collinear three-body problem with negative energy, J. Differential Equations, 119 (1995), 284-309. doi: 10.1006/jdeq.1995.1092. [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), 113-120. [6] D. G. Saari and Z. Xia, The existence of oscillatory and super hyperbolic motion in newtonian systems, J. Differential Equations, 82 (1989), 342-355. doi: 10.1016/0022-0396(89)90137-X. [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), 95-120. doi: 10.1007/s10569-007-9070-0. [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), 191-207. doi: 10.1007/s10569-008-9175-0. [9] C. Siegel and J. Moser, "Lectures on Celestial Mechanics," Springer-Verlag, New York-Heidelberg, 1971. [10] C. Simó, Masses for which triple collision is regularizable, Celestial Mech., 21 (1980), 25-36. doi: 10.1007/BF01230243. [11] H. Yoshida, A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica D, 29 (1987), 128-142. doi: 10.1016/0167-2789(87)90050-9.
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