The variational minimization solutions of 3-body problems with 2 fixed centers

In the Newtonian 3-body problem with two fixed centers in R3, two particles with equal masses are assumed fixed and the trajectory of the third particle is affected by the two fixed particles according to Newton's second law and the general gravitational law. We show that the minimization of the action functional on suitable classes of loops yields collision-free periodic orbits of the 3-body problem provided that some simple conditions on the period are satisfied; more precisely, geometric information is obtained for such variational minimization solutions.