Periodic orbits close to elliptic tori and applications to the three-body problem

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Volumn 3, Issue 1, 2004, Pages 87-138

  Berti, Massimiliano ^a   Biasco, Luca ^b   Valdinoci, Enrico ^c  

^a SISSA   (Italy)

^b ROMA TRE UNIVERSITY   (Italy)

^c UNIVERSITY OF ROME TOR VERGATA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 8844237714     PISSN: 0391173X     EISSN: 20362145     Source Type: Journal    
DOI: None     Document Type: Review

References (30)

1. A. Ambrosetti, Critical points and nonlinear variational problems, Mm. Soc. Math. Fr., Nouv. Sr. 49, 1992.
2. A. Ambrosetti - V. Coti-Zelati - I. Ekeland, Symmetry breaking in Hamiltonian systems, J. Differential Equation 67 (1987), 165-184.
3. A. Ambrosetti - M. Badiale, Homoclinics: Poincaré-Melnikov type results via a variational approac, Ann. Inst. H. Poincaré - Analyse Non Linéaire 15, n.2 (1998), 233-252.
4. V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspekhi Mat. Nauk 18 (1963), 91-192.
5. D. Bernstein - A. Katok, Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians, Invent. Math. 88 (1987), 225-241.
6. M. Berti - L. Biasco - P. Bolle, Drift in phase space, a new variational mechanism with optimal diffusion time, J. Math. Pures Appl. 82 (2003), 613-664.
7. M. Berti - P. Bolle,, A functional analysis approach to Arnold Diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 395-450.
8. L. Biasco - L. Chierchia - E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Ration. Mech. Anal. 170 (2003), 91-135.
9. G. D. Birkhoff, Une generalization á n-dimensions du dernier théorème de géometrié de Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 192 (1931), 196-198.
10. G. D. Birkhoff - D. C. Lewis, On the periodic motions near a given periodic motion of a dynamical system, Ann. Mat. Pura Appl., IV. Ser. 12 (1933), 117-133.
11. J. Bourgain, On Melnikov's persistency problem, Math. Res. Lett. 4 (1997), 445-458.
12. H. W. Broer - G. B. Huitema - M. B. Sevriuk, “Quasi periodic motions in families of dynamical systems”, Lecture Notes in Math. 1645, Springer, 1996.
13. C. Conley - E. Zehnder, “An index theory for periodic solutions of a Hamiltonian system”, Lecture Notes in Mathematics 1007, Springer, 1983, 132-145.
14. C. Conley - E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold, Invent. Math. 73 (1983), 33-49.
15. L. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 15 (1988), 115-147.
16. W. H. Jefferys, Periodic orbits in the three-dimensional three-body problem, Astronom. J. 71 (1966), 566-567.
17. W. H. Jefferys - J. Moser, Quasi-periodic solutions for the three-body problem, Astronom. J. 71 (1966), 568-578.
18. Á. Jorba - J. Villanueva, On the Normal Behaviour of Partially Elliptic Lower Dimensional Tori of Hamiltonian Systems, Nonlinearity 10 (1997), 783-822.
19. S. B. Kuksin, Perturbation theory of conditionally periodic solutions of infinite-dimensional Hamiltonian systems and its applications to the Korteweg-de Vries equation, Mat. Sb. (N.S.) 136 (178) (1988), 396-412, 431; translation in Math. USSR-Sb. 64 (1989), 397-413.
20. J. Laskar - P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian, Celestial Mech. Dynam. Astronom. 62 (1995), 193-217.
21. D. C. Lewis, Sulle oscillazioni periodiche di un sistema dinamico, Atti Accad. Naz. Rend. Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (1934), 234-237.
22. V. K. Melnikov, On certain cases of conservation of almost periodic motions with a small change of the Hamiltonian function, Dokl. Akad. Nauk SSSR 165 (1965), 1245-1248.
23. J. Moser, “Proof of a generalized form of a fixed point Theorem due to G. D. Birkhoff”, In: Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNP, Rio de Janeiro, 1976), 464-494. Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977.
24. H. Poincaré, “Les Méthodes nouvelles de la Mécanique Céleste”, Gauthier Villars, Paris, 1892.
25. J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35 (1982), 653-696.
26. J. Pöschel, On elliptic lower dimensional tori in Hamiltonian system, Math. Z. 202 (1989), 559-608.
27. J. Pöschel, A KAM-Theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), 119-148.
28. P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasiperiodic motions, Celestial Mech. Dynam. Astronom. 62 (1995), 219-261.
29. C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), 479-528.
30. J. Xu - J. You, Persistence of lower-dimensional tori under the first Melnikov's nonresonance condition, J. Math. Pures Appl. 80 (2001), 1045-1067.