Avoiding collisions under topological constraints in variational   problems coming from celestial mechanics
  In a singular potential setting, we generalize a method which allows to show that minimizers under topological constraints of the action functional (or of the Maupertuis') are collision-free. This methods applies to $3$-dimensional problems of celestial mechanics exhibiting a particular cylindrical symmetry, as well as to planar problems of $N$-centre type, where it gives optimal results. 
Nicola Soave and Susanna Terracini
math.DS