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Periodic solutions to some problems of \(n\)-body type. (English) Zbl 0782.70010
Summary: We prove the existence of at least one \(T\)-periodic solution to a dynamical system of the type (1) \(-m_ i \ddot u_ i=\sum^ n_{j=1,j \neq i} \nabla V_{ij}(u_ i-u_ j,t)\), where the potentials \(V_{ij}\) are \(T\)-periodic in \(t\) and singular at the origin, \(u_ i \in \mathbb{R}^ k\), \(i=1,\ldots,n\), and \(k \geq 3\). We also provide estimates on the \(H^ 1\) norm of this solution. The proofs are based on a variant of the Lyusternik-Shnirel’man method. The results here generalize to the \(n\)- body problem some results obtained by A. Bahri and P. H. Rabinowitz on the 3-body problem [Ann. Inst. Henri Poincaré, Anal. Non. Linéaire 8, No. 6, 561-649 (1991; Zbl 0745.34034)].

MSC:
70F10 \(n\)-body problems
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C25 Periodic solutions to ordinary differential equations
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