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Variational approach to second species periodic solutions of Poincaré of the 3 body problem

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  • We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincaré second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincaré only sketched the proof of the existence of second species solutions. Rigorous proofs appeared much later and only for the restricted 3 body problem. We develop a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem. As an application, we give a rigorous proof of the existence of a class of second species solutions.
    Mathematics Subject Classification: 70F10, 70F15, 37N05.

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  • [1]

    V. M. Alexeyev and Y. S. Osipov, Accuracy of Keplerapproximation for fly-by orbits near an attracting center, Erg. Th. & Dyn. Syst., 2 (1982), 263-300.

    [2]

    V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," Encyclopedia of Math. Sciences, vol. 3, Springer-Verlag, 1989.

    [3]

    V. I. Arnold, Small denominators and problems of stability of motionin classical and celestial mechanics, Usp. Mat. Nauk., 18 (1963), 91-193. (translation in Russian Math. Surveys, 18 (1963), 191-257).

    [4]

    E. Belbruno, "Capture Dynamics and Chaotic Motions in Celestial Mechanics," Princeton University Press, NJ, 2004.

    [5]

    G. Birkhoff, "Dynamical Systems," AMS Colloquium Publications, vol. IX, 1927.

    [6]

    S. Bolotin, Shadowing chains of collision orbits, Discrete & Contin. Dyn. Syst., 14 (2006), 235-260.doi: 10.3934/dcds.2006.14.235.

    [7]

    S. Bolotin, Second species periodic orbits of the elliptic 3 body problem, Celest. & Mech. Dynam. Astron., 93 (2006), 345-373.

    [8]

    S. Bolotin, Symbolic dynamics of almost collision orbits and skew products of symplectic maps, Nonlinearity, 19 (2006), 2041-2063.doi: 10.1088/0951-7715/19/9/003.

    [9]

    S. Bolotin and R. S. MacKay, Periodic and chaotic trajectories of the second species for the $n$-centre problem, Celest. Mech. & Dynam. Astron., 77 (2000), 49-75.doi: 10.1023/A:1008393706818.

    [10]

    S. Bolotin and P. NegriniShilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system, paper in preparation.

    [11]

    S. Bolotin and D. Treschev, Hill's formula, Uspekhi Mat. Nauk, 65 (2010), 3-70; (translation in Russian Math. Surveys, 65 (2010), 191-257.)

    [12]

    N. Fenichel, Asymptotic stability with rate conditions for dynamical systems, Bull. Am. Math. Soc., 80 (1974), 346-349.doi: 10.1090/S0002-9904-1974-13498-1.

    [13]

    J. Font, A. Nunes and C. Simo, Consecutive quasi-collisions in the planar circular RTBP, Nonlinearity, 15 (2002), 115-142.doi: 10.1088/0951-7715/15/1/306.

    [14]

    G. Gomez and M. Olle, Second species solutions in the circular and elliptic restricted three body problem, I and II, Mech. & Dynam. Astron., 52 (1991), 107-146 and 147-166.

    [15]

    J. P. Marco and L. Niederman, Sur la construction des solutions deseconde espèce dans le problème plan restrient des trois corps, Ann. Inst. H. Poincare Phys. Théor., 62 (1995), 211-249.

    [16]

    R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.doi: 10.1007/BF01941322.

    [17]

    L. M. Perko, Second species solutions with an $O(u^v)$, $0< v < 1$ near-Moon passage, Celest. Mech., 24 (1981), 155-171.doi: 10.1007/BF01229193.

    [18]

    A. Poincaré, "Les Methodes Nouvelles de la Mecanique Celeste," Volume 3. Gauthier-Villars, Paris, 1899.

    [19]

    C. Simo, Solution of Lambert's problem by means of regularization, Collect. Math., 24 (1973), 231-247.

    [20]

    L. P. Shilnikov, On a Poincaré-irkhoff problem, Math. USSR Sbornik, 3 (1967), 353-371.

    [21]

    D. V. Turaev and L. P. Shilnikov, Hamiltonian systems with homoclinic saddle curves, (Russian) Dokl. Akad. Nauk SSSR, 304 (1989), 811-814; (translation in Soviet Math.Dokl., 39 (1989), 165-168).

    [22]

    E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," Cambridge University Press, 1988.

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