  After the existence proof of the first remarkably stable simple choreographic
motion-- the figure eight of the planar three-body problem by Chenciner and
Montgomery in 2000, a great number of simple choreographic solutions have been
discovered numerically but very few of them have rigorous existence proofs and
none of them are stable. Most important to astronomy are stable periodic
solutions which might actually be seen in some stellar system. A question for
simple choreographic solutions on $n$-body problems naturally arises: Are there
any other stable simple choreographic solutions except the figure eight?
  In this paper, we prove the existence of infinitely many simple choreographic
solutions in the classical Newtonian 4-body problem by developing a new
variational method with structural prescribed boundary conditions (SPBC).
Surprisingly, a family of choreographic orbits of this type are all linearly
stable. Among the many stable simple choreographic orbits, the most
extraordinary one is the stable star pentagon choreographic solution. The star
pentagon is assembled out of four pieces of curves which are obtained by
minimizing the Lagrangian action functional over the SPBC.
  We also prove the existence of infinitely many double choreographic periodic
solutions, infinitely many non-choreographic periodic solutions and uncountably
many quasi-periodic solutions. Each type of periodic solutions have many stable
solutions and possibly infinitely many stable solutions.
