Asymptotic density of collision orbits in the restricted circular planar 3 body problem. (English) Zbl 1471.70008
In this paper, the system of Sun-Jupiter (as primaries) and Asteroid has been considered with the mass of Jupiter normalized by \(\mu\) and the Sun’s by \(1-\mu\) and the mass of Asteroid considered as zero and moves by the influence of the gravity of the primaries. It is shown that there exists an open set \(U\) in phase space of fixed measure, for the restricted planar 3-body problem, where the set of initial points that leads to collision orbits is \(O(\mu^{\frac{1}{20}})\) dense as \(\mu\) tends to zero.
Reviewer: Maria Gousidou-Koutita (Thessaloniki)
MSC:
| 70F07 | Three-body problems |
| 70F16 | Collisions in celestial mechanics, regularization |
| 37N05 | Dynamical systems in classical and celestial mechanics |
| 70F15 | Celestial mechanics |
Keywords:
restricted circular planar 3 body problem; collision orbits; phase space; Delaunay coordinates; Levi-Civita regularizationReferences:
| [1] | Alekseev, V.: Final motions in the three body problem and symbolic dynamics. Russ. Math. Surv. 36(4), 181-200 (1981) · Zbl 0503.70006 |
| [2] | Alexeyev, V.: Sur l’allure finale du mouvement dans le problème des trois corps, Actes du Congrès International des mathématiciens, t.2 (Nice, Sept. 1970), Gauthier-Villars, Paris, 893-907 1971 · Zbl 0266.70005 |
| [3] | Arnol’d, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk18(6), 91-191 1963. MR 30, 943. Russian Math. Surveys 18:6 (1963), 85-160 · Zbl 0135.42701 |
| [4] | Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Dynamical Systems III. Springer, Berlin (1988) |
| [5] | Birkhoff, G.D.: Dynamical systems. Am. Math. Soc. Colloq. Publ. vol. IX, p. 290 1966 · Zbl 0171.05402 |
| [6] | Bolotin, S., Mckay, R.: Periodic and chaotic trajectories of the second species for the \[n-\] n-center problem. Celest. Mech. Dyn. Astron. 77, 49-75 (2000) · Zbl 0979.70009 |
| [7] | Bolotin, S., Mckay, R.: Nonplanar second species periodic and chaotic trajectories for the circular restricted three-body problem. Celest. Mech. Dyn. Astron. 94(4), 433-449 (2006) · Zbl 1094.70010 |
| [8] | Bolotin, S.: Second species periodic orbits of the elliptic 3 body problem. Celest. Mech. Dyn. Astron. 93(1-4), 343-371 (2005) · Zbl 1129.70006 |
| [9] | Bolotin, S.: Symbolic dynamics of almost collision orbits and skew products of symplectic maps. Nonlinearity 19(9), 2041-2063 (2006) · Zbl 1193.70025 |
| [10] | Bolotin, S., Negrini, P.: Variational approach to second species periodic solutions of Poincaré(C) of the 3 body problem. Discret. Contin. Dyn. Syst. 33(3), 1009-1032 (2013) · Zbl 1263.70012 |
| [11] | Cassels, J.W.S.: An introduction to Diophantine Approximation. The Syndics of the Cambridge University Press, Cambridge (1957) · Zbl 0077.04801 |
| [12] | Chazy, J.: Sur l’allure finale du mouvement dans le probléme des trois corps quand le temps croit indefiniment. Ann. Sci. École Norm. Sup. 3(39), 29-130 (1922) · JFM 48.1074.04 |
| [13] | Chenciner, A., Llibre, J.: A note on the existence of invariant punctured tori in the planar circular restricted three body problem. Ergod. Theory & Dyn. Syst. 8, 63-72 (1988) · Zbl 0657.70016 |
| [14] | Chierchia, L., Pinzari, G.: The planetary N-body problem: symplectic foliation, reductions and invariant tori. Invent. Math. 186(1), 1-77 (2011) · Zbl 1316.70010 |
| [15] | Fejoz, J.: Quasi periodic motions in the planar three-body problem. J. Differ. Equ. 183(2), 303-341 (2002) · Zbl 1057.70006 |
| [16] | Fejoz, J.: Démonstration du théorème d’Arnold sur la stabilité du système planétaire (d’aprés Michael Herman). Ergod. Theory Dyn. Syst. 24(5), 1521-1582 (2004) · Zbl 1087.37506 |
| [17] | Féjoz, J., Guardia, M., Kaloshin, V., Roldán, P.: Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three body problem. J. Eur. Math. Soc. 18(10), 2315-2403 (2016) · Zbl 1404.70028 |
| [18] | Grobman, D.: Homeomorphism of systems of differential equations. Dokl. Akad. Nauk SSSR 128, 880-881 (1959) · Zbl 0100.29804 |
| [19] | Guysinsky, M., Hasselblatt, B., Rayskin, V.: Differentiability of the Hartman Grobman linearization. Discret. Contin. Dyn. Syst. 9(4), 979-984 (2003) · Zbl 1024.37022 |
| [20] | Hartman, P.: A lemma in the theory of structural stability of differential equations. Proc. Am. Math. Soc. 11, 610-620 (1960) · Zbl 0132.31904 |
| [21] | Hartman, P.: On the local linearization of differential equations. Proc. Am. Math. Soc. 14, 568-573 (1963) · Zbl 0115.29801 |
| [22] | Herman, M.: Sur les courbes invariantes par les difféomorphismes de l’anneau. Soc. Math. De France, 248 1986 · Zbl 0532.58011 |
| [23] | Herman, M.: Some open problems in dynamical systems. Proceedings of the ICM, 1998, Volume II, Doc. Math. J. DMV, pp. 797-808 · Zbl 0910.58036 |
| [24] | Knauf, A., Fleischer, S.: Improbability of Wandering Orbits Passing Through a Sequence of Poincaré Surfaces of Decreasing Size, available on arXiv:1802.08566 · Zbl 1459.37018 |
| [25] | Knauf, A., Fleischer, S.: Improbability of Collisions in n-Body Systems, available on arXiv:1802.08564 · Zbl 1448.70029 |
| [26] | Kolmogorov, A.N.: On the conservation of conditionally periodic motion under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSR98, 527-530 · Zbl 0056.31502 |
| [27] | Marco, J.P., Niederman, L.: Sur la construction des solutions de seconde espece dans le probleme plan restreint des trois corps. Ann. Inst. H. Poincaré Phys. Théor. 62(3), 211-249 (1995) · Zbl 0821.70006 |
| [28] | Moeckel, R.: Orbits of the three-body problem which pass infinitely close to triple collision. Am. J. Math. 103(6), 1323-1341 (1981) · Zbl 0475.70008 |
| [29] | Moeckel, R.: Chaotic dynamics near triple collision. Arch. Ration. Mech. Anal. 107(1), 37-69 (1989) · Zbl 0697.70021 |
| [30] | Moeckel, R.: Symbolic dynamics in the planar three-body problem. Regul. Chaot. Dyn. 12(5), 449-475 (2007) · Zbl 1229.70033 |
| [31] | Poincaré, H.: Les méthodes nouvelles de la mécanique céleste, vol. 3, pp. 1892-1894. Gauthier-Villars, Paris · JFM 30.0834.08 |
| [32] | Pöschel, J.: Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math. 35(5), 653-696 (1982) · Zbl 0542.58015 |
| [33] | Laskar, J., Robutel, P.: Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian. Celest. Mech. Dyn. Astron.62(3), 193-217 1995 · Zbl 0837.70008 |
| [34] | Robutel, P.: Stability of the planetary three-body problem. II. KAM. Celest. Mech. Dyn. Astron. 62(3), 219-261 (1995) · Zbl 0837.70009 |
| [35] | Saari, D.G.: Improbability of collisions in Newtonian gravitational systems. Trans. Am. Math. Soc. 162, 267-271 (1971) · Zbl 0231.70007 |
| [36] | Saari, D.G.: Improbability of collisions in Newtonian gravitational systems. II. Trans. Am. Math. Soc. 181, 351-368 (1973) · Zbl 0283.70007 |
| [37] | Siegel, C.L.: Vorlesungen iiber Himmelsmechanik, pp. 18-178. Springer, Berlin 1956 · Zbl 0070.45403 |
| [38] | Sitnikov, K.A.: The existence of oscillatory motions in the three-body problem. Dokl. Akad. Nauk SSSR133, 303-306 1960. MR 23 B435. Soviet Physics Dokl. 5 (1961), 647-650 · Zbl 0108.18603 |
| [39] | Zhao, L.: Quasi-periodic almost-collision orbits in the spatial three-body problem. Commun. Pure Appl. Math.LXVIII, 2144-2176 2015 · Zbl 1393.70029 |
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