New periodic solutions for planar N+2-body problems

► We use variational methods to prove the existence of non-collision periodic solutions with certain symmetric and topological constraints. ► The N+2 bodies move on two or three different closed curves, which is a new solution different from all the examples ever found. ► Some of masses have period 1, and others have period 1/N. ► When we prove that the solutions are collision-free, we use Zhang-Zhou's arguments.