×

Collision and non-collision solutions for a class of Keplerian-like dynamical systems. (English) Zbl 0832.70009

Based on one-to-one correspondence between periodic solutions of the second order Hamiltonian system \((*)\) \(q'' = - \nabla V(t,q (t))\) and the critical points of the functional \(f(u) = {1 \over 2} \int^T_0 |u'(t) |^2 dt - \int^T_0 V(t,u (t))dt\), where \(q(t) \in R^N\), \(u(t) \in H^1(S^1, R^N \backslash \{0\})\) is \(T\)- periodic in \(t\), and \(V \in C^1 (R \times R^N \backslash \{0\},R)\), \(V(t,x) \to - \infty\) as \(|x |\to 0\), the authors look for conditions on \(V\) which ensure the existence of a non-collisional (classical) solution of \((*)\). First of all, they prove that every generalized solution of an autonomous system has only finitely many collisions and then, extending this result to non-autonomous systems and using Morse index theory, provide a bound on the number of collisions. Finally, they prove the existence of a non-collisional solution of \((*)\) when \(V\) behaves near the origin like \(- |x |^\alpha\) with \(\alpha \in (1,2)\).
Reviewer: I.Mladenov (Sofia)

MSC:

70F05 Two-body problems
70H05 Hamilton’s equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ambrosetti, A.; Coti Zelati, V., Critical points with lack of compactness and singular dynamical systems, Ann. Mat. Pura Appl. (4), 149, 237-259 (1987) · Zbl 0642.58017
[2] Ambrosetti, A.; Coti Zelati, V., Perturbations of Hamiltonian systems with Keplerian potentials, Math. Z., 201, 227-242 (1989) · Zbl 0653.34032
[3] Bahri, A.; Lions, P. L., Morse index of some min-max critical point. — I. Application to multiplicity results, Comm. Pure Appl. Math., 41, 1027-1037 (1988) · Zbl 0645.58013
[4] Bahri, A.; Rabinowitz, P. H., A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal., 82, 412-428 (1989) · Zbl 0681.70018
[5] Coti Zelati, V., Periodic solutions for a class of planar, singular dynamical systems, J. Math. Pures Appl., 68, 109-119 (1989) · Zbl 0633.34034
[6] Degiovanni, M.; Giannoni, F., Periodic solutions of dynamical systems with Newtonian-type potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4, 467-494 (1989) · Zbl 0692.34050
[7] Gordon, W., Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc., 204, 113-135 (1975) · Zbl 0276.58005
[8] Greco, C., Periodic solutions of a class of singular Hamiltonian systems, Nonlinear Anal., 12, 259-269 (1988) · Zbl 0648.34048
[9] E.Serra - S.Terracini,Noncollision solutions to some singular minimzation problems with Keplerian-like potentials, Nonlinear Anal., to appear. · Zbl 0813.70006
[10] Solimini, S., Morse index estimates in min-max theorems, Manuscripta Math., 63, 421-453 (1989) · Zbl 0685.58010
[11] Tanaka, K., Morse index at critical points related to the symmetric mountain pass theorem and applications, Commun. Partial Diff. Eq., 14, 119-128 (1989) · Zbl 0669.34035
[12] S.Terracini,An homotopical index and multeplicity of periodic solutions to dynamical systems with singular potentials, J. Diff. Eq., to appear.
[13] S.Terracini,Second order conservative systems with singular potentials: non collision periodic solutions to fixed energy problem, preprint (1991).
[14] Viterbo, C., Indice de Morse des points critiques obtenus par minimax, Ann. Inst. H. Poincaré, Analyse non linéaire, 3, 221-225 (1988) · Zbl 0695.58007
[15] K.Tanaka,Non-collisions for a second order singular Hamiltonian system with weak force, preprint (1991).
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.