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

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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

Sergey V. Bolotin ^1, and Piero Negrini ^2,

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

Abstract
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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.

Keywords: collision., action functional, 3 body problem, periodic orbit.

Mathematics Subject Classification: 70F10, 70F15, 37N0.

Citation: Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete and Continuous Dynamical Systems, 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), 263-300.
[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. Negrini, Shilnikov 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.

show all references

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), 263-300.
[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. Negrini, Shilnikov 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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