Abstract
We consider the plane restricted elliptic 3 body problem with small mass ratio and small eccentricity and prove the existence of many periodic orbits shadowing chains of collision orbits of the Kepler problem. Such periodic orbits were first studied by Poincaré for the non-restricted 3 body problem. Poincaré called them second species solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. M. Alexeyev (1970) ArticleTitle‘Sur l’allure finale du mouvement dans le problème des trois corps’ Actes du Congrès Int. Math. 2 893–907
V. M. Alexeyev Y. S. Osipov (1982) ArticleTitle‘Accuracy of Kepler approxiamtion for fly-by orbits near an attracting center’ Erg. Theory and Dyn. Syst. 2 263–300 Occurrence Handle86c:70011
V. I. Arnold V. V. Kozlov A. I. Neishtadt (1989) Mathematical aspects of Classical and Celestial Mechanics, Enciklopedia of Mathematical Sciences Springer-Verlag Berlin
E. Belbruno (2004) Capture Dynamics and Chaotic Motions in Celestial Mechanics Princeton University Press Princeton
S. V. Bolotin R. S. MacKay (2000) ArticleTitle‘Periodic and chaotic trajectories of the second species for the n-centre problem’ Cel. Mech. Dyn. Astron. 77 49–75 Occurrence Handle2000CeMDA..77...49B Occurrence Handle2002a:70008
S. V. Bolotin (2006) ArticleTitleShadowing chains of collision orbits Special Issue of Discr. and Cont. Dynam. Syst. A 14 235–260
A. D. Bruno (1981) ArticleTitle‘On periodic flybus to the Moon’ Cel. Mech. Dyn. Astron. 24 255–268 Occurrence Handle0523.70017 Occurrence Handle82i:70011
A. D. Bruno (1990) Restricted 3 Body Problem Nauka Moscow
J. Font A. Nunes C. Simo (2002) ArticleTitle‘Consecutive quasi-collisions in the planar circular RTBP’ Nonlinearity 15 115–142 Occurrence Handle10.1088/0951-7715/15/1/306 Occurrence Handle2002Nonli..15..115F Occurrence Handle2002i:70011
G. Gomez M. Olle (1991) ArticleTitle‘Second species solutions in the circular and elliptic restricted three body problem, I. Existence and Asymptotic Approximation Cel. Mech. Dyn. Astron. 52 107–146 Occurrence Handle1991CeMDA..52..107G Occurrence Handle93f:70008
G. Gomez M. Olle (1991) ArticleTitle‘Second species solutions in the circular and elliptic restricted three body problem, II. Numerical Exploration’ Cel. Mech. Dyn. Astron. 52 147–166 Occurrence Handle1991CeMDA..52..147G Occurrence Handle93f:70009
M. R. Henon (1977) Generating Families in the Restricted 3 Body Problem Springer Berlin
Kozlov V. V. and Treschev, D. V.: 1991, Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts. Transl. of Math. Monographs, Vol. 89 AMS.
R. S. MacKay J. D. Meiss (1983) ArticleTitle‘Linear stability of periodic orbits in Lagrangian systems’ Phys. Lett. A 98 92–94 Occurrence Handle10.1016/0375-9601(83)90735-1 Occurrence Handle1983PhLA...98...92M Occurrence Handle85a:58031
J. P. Marco L. Niederman (1995) ArticleTitle‘Sur la construction des solutions de seconde espèce dans le problème plan restrient des trois corps’ Ann. Inst. H. Poincare Phys. Théory 62 211–249 Occurrence Handle96c:70011
J. P. Marco D. Sauzin (2004) ArticleTitle‘Wandering domains and random walks in Gevrey near integrable systems’ Erg. Theory and Dyn. Syst. 24 1619–1666 Occurrence Handle2005g:37115
Moeckel R.: 2002, ‘Generic drift on Cantor sets of annuli’, Contemp Math. AMS, 163–171.
L. M. Perko (1981) ArticleTitle‘Second species solutions with an O(μν), 0< ν < 1 near-Moon passage’ Celest. Mech. 24 155–171 Occurrence Handle1981CeMec..24..155P Occurrence Handle0478.70008 Occurrence Handle82j:70008
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bolotin, S. Second Species Periodic Orbits of the Elliptic 3 Body Problem. Celestial Mech Dyn Astr 93, 343–371 (2005). https://doi.org/10.1007/s10569-005-2172-7
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10569-005-2172-7