New styles of periodic solutions of the classical six-body problem

New styles of periodic solutions (weaving styles and chasing styles) of the classical equal mass six-body problem are shown to exist using a variational method and computational algorithm. These solutions with axial symmetry and collision-free of the bodies. We first prove existence of collision-free minimizers of the Lagrangian action functional on suitable class of symmetric loops, which yields collision-free periodic solutions of the equal mass six-body problem. Then we derive approximate analytical solutions for the new families of the weaving and chasing periodic solutions in the form of uniformly convergent series with easily computable components in the reproducing kernel Hilbert space, the derivation was performed by using the multi-step reproducing kernel Hilbert space method. Moreover, the new styles of periodic solutions are numerically illustrated.