Symmetric orbits arising from Figure-Eight for N-body problem

In this paper, we first describe how we can arrange any bodies on Figure-Eight without collision in a dense subset of $[0,T]$ after showing that the self-intersections of Figure-Eight will not happen in this subset. Then it is reasonable for us to consider the existence of generalized solutions and non-collision solutions with Mixed-symmetries or with Double-Eight constraints, arising from Figure-Eight, for N-body problem. All of the orbits we found numerically in Section ref{se7} have not been obtained by other authors as far as we know. To prove the existence of these new periodic solutions, the variational approach and critical point theory are applied to the classical N-body equations. And along the line used in this paper, one can construct other symmetric constraints on N-body problems and prove the existence of periodic solutions for them.