Periodic Solutions for N-Body-Type Problems

Abstract

The authors consider non-autonomous N-body-type problems with strong force type potentials at the origin and sub-quadratic growth at infinity. Using Ljusternik-Schnirelmann theory, the authors prove the existence of unbounded sequences of critical values for the Lagrangian action corresponding to non-collision periodic solutions.

Keywords

Periodic solutions / N-body type problems / Variational methods

References

[1] Gordon W B. Conservative dynamical system involving strong forces. Trans. Amer. Math. Soc., 1975, 204: 113-135

[2] Ambrosetti A, Coti Zelati V. Critical points with lack of copactness and applications to singular dynamical system. Ann. Mat. Pura Appl., 1987, 149: 237-259

[3] Ambrosetti A, Coti Zelati V. Periodic Solutions of Singular Lagrangian Systems, 1993, Boston: Birkhäuser

[4] Jiang M. A remark on periodic solutions of singular Hamiltonian systems. Acta Math. Sinica, 1992, 35: 758-763

[5] Majer P. Ljusternik-Schnirelmann theory with local Palais-Smale conditions and singular dynamical systems. Ann. Inst. H. Poincare Analyse non Lineaire, 1991, 8: 459-476

[6] Bahri A, Rabinowitz P. Periodic solutions of Hamiltonian systems of three body type. Ann. IHP. Nonlineaire, 1991, 8: 561-649

[7] Majer P, Terracini S. Periodic solutions to some problems of n-body type. Arch. Rational Mech. Anal., 1993, 124: 381-404

[8] Fadell E, Husseini S. Infinite cup length in free loop spaces with an application to a problem of the N-body type. Ann. Inst. H. Poincare Analyse nonLineaire, 1992, 9: 305-319

[9] Chenciner A. Action minimizing solutions of the Newtonian n-body problem: From homology to symmetry. ICM, 2002, 3: 279-294 1, 641–643

[10] Bessi U, Coti Zelati V. Symmetries and noncollision closed orbits for planar N-body type problems. Nonlinear Analysis, 1991, 16: 587-598

[11] Chen K C. Variational methods on periodic and quasi-periodic solutions for the N-body problems. Ergodic Theory and Dynamical Systems, 2003, 23: 1691-1715

[12] Chenciner A, Montgomery R. A remarkable periodic solution of the three body problem in the case of equal masses. Annals of Math., 2000, 152: 881-901

[13] Gordon W. A minimizing property of Keplerian orbits. Amer. J. Math., 1977, 5: 961-971

[14] Coti Zelati V. The periodic solutions of N-body-type problems. Ann. Inst. H. Poincare Analyse non Lineaire, 1990, 7: 477-492

[15] Degiovanni M, Giannoni F. Dynamical systems with Newtonian type potentials. Ann. Scuola Norm. Sup. Pisa., 1989, 15: 467-494

[16] Ferrario D, Terracini S. On the existence of collisionless equivariant minimizers for the classical n-body problem. Invention Math., 2004, 155: 305-362

[17] Marchal C. How the method of minimization of action avoids singularities. Cel. Mech. and Dyn. Astronomy, 2002, 83: 325-353

[18] Moore C. Braids in classical gravity. Physical Review Letters, 1993, 70: 3675-3679

[19] Simo C. Dynamical properties of the figure eight solution of the three-body problem, 2002, Providence, RI: AMS 209-228

[20] Simo C. New families of solutions in N-body problems. Progress in Math., 2001, 201: 101-115

[21] Venturelli A. Une caracterisation variationnelle des solutions de Lagrange du probleme plan des trois corps. C.R. Acad. Sci. Paris, 2001, 332: 641-644

[22] Zhang S Q, Zhou Q. A minimizing property of Lagrangian solutions. Acta Math. Sinical, 2001, 17: 497-500

[23] Zhang S Q, Zhou Q. Variational methods for the choreography solution to the three-body problem. Science in China, 2002, 45: 594-597

[24] Zhang S Q, Zhou Q. Symmetric periodic noncollision solutions for N-body-type problems. Acta Math. Sinica, New Series, 1995, 11: 37-43