# American Institute of Mathematical Sciences

March  2013, 33(3): 1009-1032. doi: 10.3934/dcds.2013.33.1009

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

 1 Department of Mathematics, University of Wisconsin, Madison, United States 2 Department of Mathematics, La Sapienza, University of Rome

Received  April 2011 Revised  February 2012 Published  October 2012

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.
Citation: Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009
##### References:
 [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.   Google Scholar [2] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,", Encyclopedia of Math. Sciences, (1989).   Google Scholar [3] V. I. Arnold, Small denominators and problems of stability of motionin classical and celestial mechanics,, Usp. Mat. Nauk., 18 (1963), 91.   Google Scholar [4] E. Belbruno, "Capture Dynamics and Chaotic Motions in Celestial Mechanics,", Princeton University Press, (2004).   Google Scholar [5] G. Birkhoff, "Dynamical Systems,", AMS Colloquium Publications, (1927).   Google Scholar [6] S. Bolotin, Shadowing chains of collision orbits,, Discrete & Contin. Dyn. Syst., 14 (2006), 235.  doi: 10.3934/dcds.2006.14.235.  Google Scholar [7] S. Bolotin, Second species periodic orbits of the elliptic 3 body problem,, Celest. & Mech. Dynam. Astron., 93 (2006), 345.   Google Scholar [8] S. Bolotin, Symbolic dynamics of almost collision orbits and skew products of symplectic maps,, Nonlinearity, 19 (2006), 2041.  doi: 10.1088/0951-7715/19/9/003.  Google Scholar [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.  doi: 10.1023/A:1008393706818.  Google Scholar [10] S. Bolotin and P. Negrini, Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system,, paper in preparation., ().   Google Scholar [11] S. Bolotin and D. Treschev, Hill's formula,, Uspekhi Mat. Nauk, 65 (2010), 3.   Google Scholar [12] N. Fenichel, Asymptotic stability with rate conditions for dynamical systems,, Bull. Am. Math. Soc., 80 (1974), 346.  doi: 10.1090/S0002-9904-1974-13498-1.  Google Scholar [13] J. Font, A. Nunes and C. Simo, Consecutive quasi-collisions in the planar circular RTBP,, Nonlinearity, 15 (2002), 115.  doi: 10.1088/0951-7715/15/1/306.  Google Scholar [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.   Google Scholar [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.   Google Scholar [16] R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19.  doi: 10.1007/BF01941322.  Google Scholar [17] L. M. Perko, Second species solutions with an $O(u^v)$, $0< v < 1$ near-Moon passage,, Celest. Mech., 24 (1981), 155.  doi: 10.1007/BF01229193.  Google Scholar [18] A. Poincaré, "Les Methodes Nouvelles de la Mecanique Celeste,", Volume 3. Gauthier-Villars, (1899).   Google Scholar [19] C. Simo, Solution of Lambert's problem by means of regularization,, Collect. Math., 24 (1973), 231.   Google Scholar [20] L. P. Shilnikov, On a Poincaré-irkhoff problem,, Math. USSR Sbornik, 3 (1967), 353.   Google Scholar [21] D. V. Turaev and L. P. Shilnikov, Hamiltonian systems with homoclinic saddle curves,, (Russian) Dokl. Akad. Nauk SSSR, 304 (1989), 811.   Google Scholar [22] E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", Cambridge University Press, (1988).   Google Scholar

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##### References:
 [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.   Google Scholar [2] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,", Encyclopedia of Math. Sciences, (1989).   Google Scholar [3] V. I. Arnold, Small denominators and problems of stability of motionin classical and celestial mechanics,, Usp. Mat. Nauk., 18 (1963), 91.   Google Scholar [4] E. Belbruno, "Capture Dynamics and Chaotic Motions in Celestial Mechanics,", Princeton University Press, (2004).   Google Scholar [5] G. Birkhoff, "Dynamical Systems,", AMS Colloquium Publications, (1927).   Google Scholar [6] S. Bolotin, Shadowing chains of collision orbits,, Discrete & Contin. Dyn. Syst., 14 (2006), 235.  doi: 10.3934/dcds.2006.14.235.  Google Scholar [7] S. Bolotin, Second species periodic orbits of the elliptic 3 body problem,, Celest. & Mech. Dynam. Astron., 93 (2006), 345.   Google Scholar [8] S. Bolotin, Symbolic dynamics of almost collision orbits and skew products of symplectic maps,, Nonlinearity, 19 (2006), 2041.  doi: 10.1088/0951-7715/19/9/003.  Google Scholar [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.  doi: 10.1023/A:1008393706818.  Google Scholar [10] S. Bolotin and P. Negrini, Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system,, paper in preparation., ().   Google Scholar [11] S. Bolotin and D. Treschev, Hill's formula,, Uspekhi Mat. Nauk, 65 (2010), 3.   Google Scholar [12] N. Fenichel, Asymptotic stability with rate conditions for dynamical systems,, Bull. Am. Math. Soc., 80 (1974), 346.  doi: 10.1090/S0002-9904-1974-13498-1.  Google Scholar [13] J. Font, A. Nunes and C. Simo, Consecutive quasi-collisions in the planar circular RTBP,, Nonlinearity, 15 (2002), 115.  doi: 10.1088/0951-7715/15/1/306.  Google Scholar [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.   Google Scholar [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.   Google Scholar [16] R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19.  doi: 10.1007/BF01941322.  Google Scholar [17] L. M. Perko, Second species solutions with an $O(u^v)$, $0< v < 1$ near-Moon passage,, Celest. Mech., 24 (1981), 155.  doi: 10.1007/BF01229193.  Google Scholar [18] A. Poincaré, "Les Methodes Nouvelles de la Mecanique Celeste,", Volume 3. Gauthier-Villars, (1899).   Google Scholar [19] C. Simo, Solution of Lambert's problem by means of regularization,, Collect. Math., 24 (1973), 231.   Google Scholar [20] L. P. Shilnikov, On a Poincaré-irkhoff problem,, Math. USSR Sbornik, 3 (1967), 353.   Google Scholar [21] D. V. Turaev and L. P. Shilnikov, Hamiltonian systems with homoclinic saddle curves,, (Russian) Dokl. Akad. Nauk SSSR, 304 (1989), 811.   Google Scholar [22] E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", Cambridge University Press, (1988).   Google Scholar
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