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.

[2]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,", Encyclopedia of Math. Sciences, (1989).

[3]

V. I. Arnold, Small denominators and problems of stability of motionin classical and celestial mechanics,, Usp. Mat. Nauk., 18 (1963), 91.

[4]

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

[5]

G. Birkhoff, "Dynamical Systems,", AMS Colloquium Publications, (1927).

[6]

S. Bolotin, Shadowing chains of collision orbits,, Discrete & Contin. Dyn. Syst., 14 (2006), 235. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. 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. doi: 10.1007/BF01229193.

[18]

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

[19]

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

[20]

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

[21]

D. V. Turaev and L. P. Shilnikov, Hamiltonian systems with homoclinic saddle curves,, (Russian) Dokl. Akad. Nauk SSSR, 304 (1989), 811.

[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.

[2]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,", Encyclopedia of Math. Sciences, (1989).

[3]

V. I. Arnold, Small denominators and problems of stability of motionin classical and celestial mechanics,, Usp. Mat. Nauk., 18 (1963), 91.

[4]

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

[5]

G. Birkhoff, "Dynamical Systems,", AMS Colloquium Publications, (1927).

[6]

S. Bolotin, Shadowing chains of collision orbits,, Discrete & Contin. Dyn. Syst., 14 (2006), 235. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. 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. doi: 10.1007/BF01229193.

[18]

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

[19]

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

[20]

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

[21]

D. V. Turaev and L. P. Shilnikov, Hamiltonian systems with homoclinic saddle curves,, (Russian) Dokl. Akad. Nauk SSSR, 304 (1989), 811.

[22]

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

[1]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080

[2]

Giovanni F. Gronchi, Chiara Tardioli. The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1323-1344. doi: 10.3934/dcdsb.2013.18.1323

[3]

Gianni Arioli. Branches of periodic orbits for the planar restricted 3-body problem. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 745-755. doi: 10.3934/dcds.2004.11.745

[4]

Elbaz I. Abouelmagd, Juan Luis García Guirao, Jaume Llibre. Periodic orbits for the perturbed planar circular restricted 3–body problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1007-1020. doi: 10.3934/dcdsb.2019003

[5]

Mark Lewis, Daniel Offin, Pietro-Luciano Buono, Mitchell Kovacic. Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137

[6]

Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158

[7]

Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85

[8]

Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062

[9]

Samuel R. Kaplan, Mark Levi, Richard Montgomery. Making the moon reverse its orbit, or, stuttering in the planar three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 569-595. doi: 10.3934/dcdsb.2008.10.569

[10]

Qunyao Yin, Shiqing Zhang. New periodic solutions for the circular restricted 3-body and 4-body problems. Communications on Pure & Applied Analysis, 2010, 9 (1) : 249-260. doi: 10.3934/cpaa.2010.9.249

[11]

Martha Alvarez-Ramírez, Joaquín Delgado. Blow up of the isosceles 3--body problem with an infinitesimal mass. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1149-1173. doi: 10.3934/dcds.2003.9.1149

[12]

Samuel R. Kaplan, Ernesto A. Lacomba, Jaume Llibre. Symbolic dynamics of the elliptic rectilinear restricted 3--body problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 541-555. doi: 10.3934/dcdss.2008.1.541

[13]

Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523

[14]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229

[15]

Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379

[16]

Alain Chenciner, Jacques Féjoz. The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 421-438. doi: 10.3934/dcdsb.2008.10.421

[17]

Jungsoo Kang. Survival of infinitely many critical points for the Rabinowitz action functional. Journal of Modern Dynamics, 2010, 4 (4) : 733-739. doi: 10.3934/jmd.2010.4.733

[18]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

[19]

Duokui Yan, Tiancheng Ouyang, Zhifu Xie. Classification of periodic orbits in the planar equal-mass four-body problem. Conference Publications, 2015, 2015 (special) : 1115-1124. doi: 10.3934/proc.2015.1115

[20]

Abimael Bengochea, Manuel Falconi, Ernesto Pérez-Chavela. Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]