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A class of periodic solutions of the $$N$$-body problem. (English) Zbl 0684.70006
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 $$k_ 2,\ldots,k_ N$$.

##### MSC:
 70F10 $$n$$-body problems 37G99 Local and nonlocal bifurcation theory for dynamical systems 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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##### References:
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