Since Poincare the crucial role of periodic orbits in the study on dynamical systems has been known.
The restricted three-body problem has intriguing dynamics which has chaotic features.  It is therefore extremely difficult to detect a periodic orbit in this problem via a direct method.  It was Poincare who first looked at periodic orbits in the rotating Kepler problem, which is integrable, and followed the family they generate into the restricted three-body problem.   Since periodic orbitws in integrable systems are easier to fine, Poincare's approach is very effective in the search of periodic orbits in the restricted three-body problem.  In this talk we will discuss the following slightly different question: Can one decide if two periodic orbits generate the same or different families is also a difficult problem? The Cieliebak-Frauenfelder-van Koert invariants, based on the Arnold $$extract_itex$$j^+$$/extract_itex$$-invariant, provide obstructions to the existence between given two periodic orbits and enable us to distinguish families.   I will give a mild introduction to these invariants and present some results on the nonexistence of certain families.