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Horseshoe periodic orbits with one symmetry in the general planar three-body problem
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 |
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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