Coti Zelati, Vittorio Periodic solutions for N-body type problems. (English) Zbl 0723.70010 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 5, 477-492 (1990). The author proves the existence of periodic solutions of assigned period T for the following system of ordinary differential equations \[ \max_ i''=\nabla_{xi}j\sum_{i\neq j}V_{ij}(x_ i-x_ j)\equiv \nabla_{xi}V. \] The method is based on minimization of a suitable function, with additional restriction on V \[ -\frac{a}{2}\sum \frac{m_ im_ j}{| x_ i-x_ j|^{\alpha}}\leq V(x_ 1,...,X_ N)\leq -\frac{b}{2}\sum \frac{m_ im_ j}{| x_ i-x_ j|^{\alpha}}, \] summations are from \(1\leq i\neq j<N\), and \(1<\frac{a}{b}<\mu\), \(\mu >1\) and depends on \(\alpha\) and the \(m_ i's\). It is further shown that the solution obtained is not a simultaneous collision one. Reviewer: N.D.Sengupta (Bombay) Cited in 1 ReviewCited in 37 Documents MSC: 70F10 \(n\)-body problems 34C25 Periodic solutions to ordinary differential equations Keywords:non-collision solution; generalized solution; existence of periodic solutions PDFBibTeX XMLCite \textit{V. Coti Zelati}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 5, 477--492 (1990; Zbl 0723.70010) Full Text: DOI Numdam EuDML References: [1] Ambrosetti, A.; Coti Zelati, V., Critical points with lack of compactness and singular dynamical systems, Ann. Mat. Pura Appl., Vol. 149, 237-259, (1987) · Zbl 0642.58017 [2] Ambrosetti, A.; Coti Zelati, V., Perturbation of Hamiltonian systems with Keplerian potentials, Math. Z., Vol. 201, 227-242, (1989) · Zbl 0653.34032 [3] Bahri, A.; Rabinowitz, P. H., A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal., Vol. 82, 412-428, (1989) · Zbl 0681.70018 [4] A. Bahri and P. H. Rabinowitz, {\it Solutions of the three-body problem via critical points at infinity}, Preprint Univ. of Wisconsin-Madison, 1988. [5] Coti Zelati, V., A class of periodic solutions of the N-body problem, Celestial Mech., Vol. 46, 177-186, (1989) · Zbl 0684.70006 [6] Degiovanni, M.; Giannoni, F., Periodic solutions of dynamical systems with Newtonian-type potentials, Ann. Sci. Norm. Sup. Pisa, Vol. 15, 467-494, (1988) · Zbl 0692.34050 [7] Gordon, W., Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc., Vol. 204, 113-135, (1975) · Zbl 0276.58005 [8] Greco, C., Periodic solutions of a class of singular Hamiltonian systems, Nonlin. Anal. TMA, Vol. 12, 259-269, (1988) · Zbl 0648.34048 [9] Rabinowitz, P. H., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., Vol. 31, 157-184, (1978) · Zbl 0358.70014 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.