Periodic orbits of -body type problems: the fixed period case

Author:
Hasna Riahi

Journal:
Trans. Amer. Math. Soc. **347** (1995), 4663-4685

MSC:
Primary 58F22; Secondary 34C25, 58E05, 70F10

DOI:
https://doi.org/10.1090/S0002-9947-1995-1316861-0

MathSciNet review:
1316861

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives a proof of the existence and multiplicity of periodic solutions to Hamiltonian systems of the form

**[1]**A. Ambrosetti and V. Coti Zelati,*Critical points with lack of compactness and applications to singular dynamical systems*, Ann. Mat. Pura Appl. (4)**149**(1987), 237-259. MR**932787 (89e:58023)****[2]**-,*Perturbation of Hamiltonian systems with Keplerian potentials*, Math. Z.**201**(1989), 227-242. MR**997224 (90g:58105)****[3]**A. Bahri, Thèse de Doctorat d'Etat, Univ. P. et M. Curie, Paris, 1981.**[4]**A. Bahri and H. Berestycki,*Forced vibrations of superquadratic Hamiltonian systems*, Acta Math.**152**(1984), 143-197. MR**741053 (85g:34041)****[5]**A. Bahri and P.H. Rabinowitz,*A minimax method for a class of Hamiltonian systems with singular potentials*, J. Funct. Anal.**82**(1989), 412-428. MR**987301 (90g:58030)****[6]**-,*Periodic solutions of Hamiltonian systems of**-body type*, Ann. Inst. H. Poincaré, Analyse, Non Lineaire**8**(1991), 561-649. MR**1145561 (92k:58223)****[7]**V. Coti Zelati,*Periodic solutions for**-body type problems*, Ann. Inst. H. Poincare, Analyse Non Linéaire**7**(1990), 477-492. MR**1138534 (93a:70009)****[8]**M. Degiovanni and F. Giannoni,*Dynamical systems with Newtonian type potentials*, Ann. Sci. Norm. Sup. Pisa**15**(1988), 467-494. MR**1015804 (91b:58201)****[9]**A. Dold,*Lectures on algebraic topology*, Springer-Verlag, 1980. MR**606196 (82c:55001)****[10]**E. Fadell and S. Husseini,*On the growth properties of the homology of free loop spaces on configuration spaces*, Proc. Conf. on Non Linear Analysis, Florida, 1992.**[11]**W.B. Gordon,*Conservative dynamical systems involving strong forces*, Trans. Amer. Math. Soc.**204**(1975), 113-135. MR**0377983 (51:14152)****[12]**C. Greco,*Periodic solutions of a class of singular Hamiltonian systems*, Nonlinear Analysis, T.M.A.**12**(1988), 259-270. MR**928560 (89d:58040)****[13]**O. Hanner,*Some theorems on absolute neighborhood retracts*, Ark. Mat.**1**(1952), 389-408. MR**0043459 (13:266d)****[14]**M.W. Hirsch,*Differential topology*, Springer-Verlag, 1988. MR**0448362 (56:6669)****[15]**A. Marino and G. Prodi,*Metodi perturbativi nella teoria di Morse*, Boll. Un. Mat. Ital.**11**(1975), 1-32. MR**0418150 (54:6192)****[16]**P. Majer and S. Terracini,*Periodic solutions to some problems of**-body type*, Arch. Rational Mech. Anal.**124**(1993), 381-404. MR**1240581 (95b:58124)****[17]**-,*Multiple periodic solutions to some**-body type problems via a collision index*, to appear.**[18]**H. Poincaré,*Les méthodes nouvelles de la mécanique céleste*, Libr. Albert Blanchard, Paris, 1987.**[19]**H. Riahi,*Study of the critical points at infinity arising from the failure of the Palais-Smale condition for**-body type problems*, Mem. Amer. Math. Soc. (to appear). MR**1445492 (99i:58028)****[20]**M. Vigué-Poirrier,*Homotopie rationnelle et croissance du nombre de géodésiques fermées*, Ann. Sci. Ecole Norm. Sup.**17**(1984), 413-431. MR**777376 (86h:58027)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1316861-0

Keywords:
-body problems,
periodic solutions,
generalized Morse inequalities

Article copyright:
© Copyright 1995
American Mathematical Society