Abstract. In this paper, we prove the existence of a family of new noncollision periodic solutions for the classical Newtonian n-body problems.In our assumption, the n = 2l ≥ 4 particles are invariant under the dihedral rotation group D l in R 3 such that, at each instant, the n particles form two twisted l-regular polygons. Our approach is variational minimizing method and we show that the minimizers are collision-free by level estimates and local deformations.

This paper is concerned with nested polygonal central configurations for the Newtonian 2n-body problem. We show that when two nested regular n-polygons ( n ≥ 3 n\geq 3 ) with masses located at the vertices form a central configuration where the twisted angle θ \theta is zero, then the value of masses in each separate polygon must be equal.

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In 1985, Perko and Walter showed that N 4 masses positioned at the vertices of a regular polygon forms a central configuration if and only if the masses are equal for the classical Newtonian N-body problem. In this paper, we extend their result to the general potential of homogeneous degree −α, where α > 0.

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