Closed orbits of fixed energy for singular Hamiltonian systems. (English) Zbl 0737.70008

The paper studies the classical \(N\)-dimensional Newtonian equations of motion in a potential field on an energy surface. The assumptions are that the potential is \(C^2\) away from the origin where it has a singularity. If the potential behaves like \(-|x|^{-a}\) with \(a>2\) the existence of periodic solutions without collisions is proved. If the potential behaves like \(-|x| ^{-b}\) with \(0<b<2\) the existence of periodic solutions that can pass through the collision is proved.


70H05 Hamilton’s equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
Full Text: DOI


[1] A. Ambrosetti,Esistenza di infinite soluzioni per problemi non lineari in assenza di parametro, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)52 (1972), 402–409. · Zbl 0249.35030
[2] A. Ambrosetti &V. Coti Zelati,Critical points with lack of compactness and singular dynamical systems, Ann. Mat. Pura e Appl. (3)149 (1987), 237–259. · Zbl 0642.58017
[3] A. Ambrosetti &G. Mancini,Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann.255 (1981), 405–421. · Zbl 0466.70022
[4] A. Bahri &P. H. Rabinowitz,A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal.82 (1989), 412–428. · Zbl 0681.70018
[5] V. Benci &F. Giannoni,Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials, J Differential Equations82 (1989), 60–70. · Zbl 0689.34034
[6] V. Coti Zelati,Dynamical systems with effective-like potentials, Nonlinear Anal. TMA12 (1988), 209–222. · Zbl 0648.34050
[7] M. Degiovanni &F. Giannoni,Periodic solutions of dynamical systems with Newtonian type potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)15 (1988), 467–494. · Zbl 0692.34050
[8] M. Degiovanni, F. Giannoni &A. Marino,Dynamical systems with Newtonian tupe potentials, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)81 (1987), 271–278. · Zbl 0667.70010
[9] E. Fadell & S. Husseini,On the category of free loop space, preprint, University of Wisconsin – Madison. · Zbl 0676.58019
[10] W. B. Gordon,Conservative dynamical systems involving strong forces, Trans. Amer. Mat. Soc.204 (1975), 113–135. · Zbl 0276.58005
[11] C. Greco,Periodic solutions of a class of singular Hamiltonian systems, Nonlinear Analysis T.M.A.12 (1988), 259–270. · Zbl 0648.34048
[12] C. Greco,Remarks on periodic solutions, with prescribed energy, for singular Hamiltonian systems, Comment. Math. Univ. Carolin.28 (1987), 661–672. · Zbl 0678.34052
[13] J. A. Hempel,Multiple solutions for a class of nonlinear boundary value problem, Indiana Univ. Math.J. 30 (1971), 983–996. · Zbl 0225.35045
[14] P. Majer,Ljusternik-Schnirelman theory with local Palais–Smale condition and singular dynamical systems, preprint, Scuola Normale Superiore – Pisa. · Zbl 0749.58046
[15] J. Moser,Regularization of Kepler’s problem and the averaging method on a manifold, Comm. Pure Appl. Math.23 (1970), 609–636. · Zbl 0193.53803
[16] Z. Nehari,Characteristic values associated with a class of nonlinear second-order differential equations, Acta Math.105 (1961), 141–175. · Zbl 0099.29104
[17] A. Szulkin,Ljusternik-Schnirelmann theory on C 1 manifolds, Ann. Inst. H. Poincaré. Anal. Non Linéaire5 (1988), 119–139. · Zbl 0661.58009
[18] S. Terracini,An homotopical index and multiplicity of periodic solutions to dynamical system with singular potential, preprint CEREMADE – Paris.
[19] E. W. C. Van Groesen,Analytical mini-max methods for Hamiltonian break orbits of prescribed energy, J. Math. Anal. Appl.132 (1988), 1–12. · Zbl 0665.70022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.