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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

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 & 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.  Google Scholar

[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. Google Scholar

[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).  Google Scholar

[4]

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

[5]

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

[6]

S. Bolotin, Shadowing chains of collision orbits, Discrete & Contin. Dyn. Syst., 14 (2006), 235-260. 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), 345-373.  Google Scholar

[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.  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), 49-75. 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), 3-70; (translation in Russian Math. Surveys, 65 (2010), 191-257.)  Google Scholar

[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.  Google Scholar

[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.  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), 107-146 and 147-166. 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), 211-249.  Google Scholar

[16]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. 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), 155-171. doi: 10.1007/BF01229193.  Google Scholar

[18]

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

[19]

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

[20]

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

[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).  Google Scholar

[22]

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

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.  Google Scholar

[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. Google Scholar

[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).  Google Scholar

[4]

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

[5]

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

[6]

S. Bolotin, Shadowing chains of collision orbits, Discrete & Contin. Dyn. Syst., 14 (2006), 235-260. 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), 345-373.  Google Scholar

[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.  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), 49-75. 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), 3-70; (translation in Russian Math. Surveys, 65 (2010), 191-257.)  Google Scholar

[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.  Google Scholar

[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.  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), 107-146 and 147-166. 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), 211-249.  Google Scholar

[16]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. 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), 155-171. doi: 10.1007/BF01229193.  Google Scholar

[18]

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

[19]

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

[20]

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

[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).  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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