Coti Zelati, Vittorio A class of periodic solutions of the \(N\)-body problem. (English) Zbl 0684.70006 Celest. Mech. Dyn. Astron. 46, No. 2, 177-186 (1989). Summary: We prove existence and multiplicity of \(T\)-periodic solutions (for any given \(T\)) for the \(N\)-body problem in \(\mathbb R^m\) (any \(m\geq 2)\) where one of the bodies has mass equal to 1 and the others have masses \(\varepsilon \alpha_ 2,\ldots,\epsilon \alpha_ N\), \(\varepsilon\) small. We find solutions such that the body of mass 1 moves close to \(x=0\) while the body of mass \(\varepsilon \alpha_ i\) moves close to one of the circular solutions of the two body problem of period \(T/k_ i\), where \(k_ i\) is any odd number. No relation has to be satisfied by \(k_ 2,\ldots,k_ N\). Cited in 19 Documents MSC: 70F10 \(n\)-body problems 37G99 Local and nonlocal bifurcation theory for dynamical systems 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Keywords:small parameter method; multiplicity of T-periodic solutions; N-body problem; circular solutions PDF BibTeX XML Cite \textit{V. Coti Zelati}, Celest. Mech. Dyn. Astron. 46, No. 2, 177--186 (1989; Zbl 0684.70006) Full Text: DOI References: [1] Ambrosetti, A.; Zelati, V. Coti; Ekeland, I., Symmetry breaking in Hamiltonian systems, J. Dif: Eq., 67, 165-184, (1987) · Zbl 0606.58043 [2] A. Ambrosetti and V. Coti Zelati: ‘Perturbation of Hamiltonian systems with Keplerian potentials’, Mathematische Zeitschrift, in print. · Zbl 0653.34032 [3] A. Ambrosetti and V. Coti Zelati: 1988, ‘Oscillations non-linéaires pour des problémes de la Mécanique Céleste’, C. R. Acad. Sci. Paris307, Série I, 569-571. · Zbl 0653.70013 [4] Gordon, W., A minimizing property of Keplerian orbits, Am. J. Math., 99, 961-971, (1977) · Zbl 0378.58006 [5] Meyer, K. R., Periodic solutions of the \(N\)-body problem, J. Diff. Eq., 39, 2-38, (1981) · Zbl 0431.70021 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.