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 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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