×

Non-integrability of the three-body problem. (English) Zbl 1270.70031

Summary: We consider the planar problem of three bodies which attract mutually with the force proportional to a certain negative integer power of the distance between the bodies. We show that such generalisation of the gravitational three-body problem is not integrable in the Liouville sense.

MSC:

70F07 Three-body problems
70H07 Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Boucher, D., Weil, J.-A.: Application of J.-J. Morales and J.-P. Ramis’ theorem to test the non-complete integrability of the planar three-body problem. In: Fauvet, F., et al. (eds.) From combinatorics to dynamical systems. Journées de calcul formel en l’honneur de Jean Thomann, Marseille, France, March 22–23, 2002. Berlin: de Gruyter. IRMA Lect. Math. Theor. Phys. 3, 163–177 (2003)
[2] Duval G., Maciejewski A.J.: Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials. Annales de l’Institut Fourier 59(7), 2839–2890 (2009) · Zbl 1196.37096 · doi:10.5802/aif.2510
[3] Julliard Tosel E.: Meromorphic parametric non-integrability; the inverse square potential. Arch. Ration. Mech. Anal. 152(3), 187–205 (2000) · Zbl 0963.70010 · doi:10.1007/PL00004233
[4] Maciejewski A.J., Przybylska M.: Partial integrability of Hamiltonian systems with homogeneous potentials. Regul. Chaotic Dyn. 15(4–5), 551–563 (2010) · Zbl 1258.70021 · doi:10.1134/S1560354710040106
[5] Morales-Ruiz J.J., Ramis J.P.: A note on the non-integrability of some Hamiltonian systems with a homogeneous potential. Methods Appl. Anal. 8(1), 113–120 (2001) · Zbl 1140.37353
[6] Morales-Ruiz J.J., Simon S.: On the meromorphic non-integrability of some N-body problems. Discret. Contin. Dyn. Syst. 24(4), 1225–1273 (2009) · Zbl 1167.37345 · doi:10.3934/dcds.2009.24.1225
[7] Simon, S.: On the meromorphic non-integrability of some problems in celestial mechanics. Ph.D. thesis, Universitat de Barcelona, Spain (2007)
[8] Tsygvintsev A.: La non-intégrabilité méromorphe du problème plan des trois corps. C. R. Acad. Sci. Paris Sér. I Math. 331(3), 241–244 (2000) · Zbl 0964.70011 · doi:10.1016/S0764-4442(00)01623-2
[9] Tsygvintsev A.: The meromorphic non-integrability of the three-body problem. J. Reine Angew. Math. 537, 127–149 (2001) · Zbl 1001.70006
[10] Tsygvintsev A.: On some exceptional cases in the integrability of the three-body problem. Celest. Mech. Dyn. Astron. 99(1), 23–29 (2007) · Zbl 1162.70313 · doi:10.1007/s10569-007-9086-5
[11] Whittaker E.T.: A Treatise on the Analytical Dynamics of Particle and Rigid Bodies with an Introduction to the Problem of Three Bodies. Cambridge University Press, London (1965)
[12] Yoshida H.: A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential. Phys. D 29(1–2), 128–142 (1987) · Zbl 0659.70012 · doi:10.1016/0167-2789(87)90050-9
[13] Yoshida H.: Justification of Painlevé analysis for Hamiltonian systems by differential Galois theory. Phys. A 288(1–4), 424–430 (2000) · doi:10.1016/S0378-4371(00)00440-4
[14] Ziglin S.L.: On involutive integrals of groups of linear symplectic transformations and natural mechanical systems with homogeneous potential. Funktsional. Anal. i Prilozhen. 34(3), 26–36 (2000) · Zbl 0995.37042 · doi:10.4213/faa309
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.