zbMATH — the first resource for mathematics

Non-collision orbits for a class of Keplerian-like potentials. (English) Zbl 0667.58055
The authors consider the existence problem of T-periodic solutions x(t), x(t)\(\neq 0\), \(\forall t\in [0,T]\) (such solutions are called non- collision solutions) of the differential equation -ẍ\(=V'(x)\), where \(V\in C^ 1({\mathbb{R}}^ n\setminus \{0\},{\mathbb{R}})\) and V(x)\(\to -\infty\) as \(| x| \to 0\). The main result is the following. If there is an open, bounded set \(\Omega \subset {\mathbb{R}}^ n\) with smooth boundary \(\Gamma\) such that: (i) \(0\in \Omega\) and is star-shaped with respect to 0; (ii) max\(\{\) V(x): \(x\in {\mathbb{R}}^ n\setminus \{0\}\}\equiv V(\xi)\), \(\forall \xi \in \Gamma\); (iii) lim sup V(x)\(<V(\Gamma)\), then for sufficiently large T there is at least one non-collision solution x such that \(\{\) x(t)\(\}\) \(\not\subset \Gamma\). For an announcement see [the authors’ paper in C. R. Acad. Sci., Paris, Sér. I 305, 813-815 (1987; Zbl 0639.34038)].

37G99 Local and nonlocal bifurcation theory for dynamical systems
Full Text: DOI Numdam EuDML
[1] Ambrosetfi, A.; Zelati, V. Coti, Solutions périodiques sans collision pour une classe de potentiels de type keplerien, C.R. Acad. Sci. Paris, 305, 813-815, (1987) · Zbl 0639.34038
[2] A. Ambrosetti and V. Con Zelati, Critical Points with Lack of Compactness and Singular Dynamical Systems, Annali di Matematica Pura ed Applicata (to appear). · Zbl 0642.58017
[3] Ambrosetti, A.; Zelati, V. Coti, Periodic solutions of singular dynamical systems, Proceed. NATO ARW “Periodic Solutions of Hamiltonian Systems and Related Topics”, (1986), Il Ciocco Italy, (to appear) · Zbl 0757.70007
[4] Capozzi, A.; Greco, C.; Salvatore, A., Lagrangian systems in presence of singularities, (1985), preprint Universitá di Bari · Zbl 0664.34054
[5] Zelati, C. Coti, Remarks on dynamical systems with weak-forces, Manuscripta Math., Vol. 57, 417-424, (1987) · Zbl 0606.58039
[6] Degiovanni, M.; Giannoni, F.; Marino, A., Dynamical systems with Newtonian potentials, Proceed. NATO ARW “Periodic Solutions of Hamiltonian Systems and Related Topics”, (1986), Il Ciocco Italy, (to appear)
[7] Gordon, W., Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc., Vol. 204, 113-135, (1975) · Zbl 0276.58005
[8] Gordon, W.; Minimizing, A., Property of Keplerian orbits, Am. Journ. of Math., Vol. 99, 961-971, (1977) · Zbl 0378.58006
[9] Greco, C., Periodic solutions of a class of singular Hamiltonian systems, (1986), Università di Bari, preprint
[10] Klingenberg, W., Lectures on closed geodesics, (1978), Springer-Verlag Berlin · Zbl 0397.58018
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.