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Periodic solutions for N-body type problems. (English) Zbl 0723.70010

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.

MSC:

70F10 \(n\)-body problems
34C25 Periodic solutions to ordinary differential equations
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References:

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