A Symmetric Spatial Periodic Orbit in the 2n-Body Problem

For each given n ≥ 2 , we study the variational existence of a new spatial periodic orbit in the 2 n -body problem. Besides excluding possible collision singularities, the main challenges left are to show that the orbit is not a relative equilibrium and it is spatial. By introducing a new estimate in the collinear central configuration, we can prove that it is not a relative equilibrium in the 2 n -body problem. Furthermore, with the help of numerical calculation, the orbit is spatial in the equal-mass four-body problem.