For planar Newtonian 4-body problems with equal masses, we use variational methods to prove the existence of a non-collision periodic choreography solution such that all bodies move on a rose-type curve with three petals. Keywords4-body problems with Newtonian potentials, rose solutions with three petals, winding numbers, variational minimization methods MSC(2000): 34C15, 34C25, 58E05Citation: Deng C H, Zhang S Q, Zhou Q. Rose solutions with three petals for planar 4-body problems.

In this paper, we consider the problem: given a symmetric concave configuration of four bodies, under what conditions is it possible to choose positive masses which make it central. We show that there are some regions in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the union of these regions, it is always possible to choose positive masses to make the configuration central.

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We study the spatial central configuration formed by two twisted regular N -polygons with any twist angle θ, and prove that the sizes of the two regular N -polygons must be equal to each other.