New periodic orbits in the planar equal-mass five-body problem

We apply our variational method to search for new periodic orbits in the planar equal-mass five-body problem. Three types of configurations are considered: all masses on a line, three masses on a line with the other two masses symmetrically locating on its two sides, and one mass on a line with the other four masses symmetrically sitting on two sides of the line. By setting suitable free variables in each pair of the three configurations and minimizing the Lagrangian action over these free variables, many new orbits were found. These orbits can be classified into 5 categories.