Periodic solutions for planar four-body problems

For Newtonian four-body problems with equal masses, we use variational minimizing methods to prove the existence of non-collision periodic solutions such that all bodies move on two symmetric trajectories with opposite orientation; here the methods and results are simpler and more general than those of Chen [K.-C. Chen, Action minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal. 158 (2001) 293–318].