A Hamiltonian version of a result of Gromoll and Grove
  The theorem that if all geodesics of a Riemannian two-sphere are closed they are also simple closed is generalized to real Hamiltonian structures on $\mathbb{R}P^3$. For reversible Finsler $2$-spheres all of whose geodesics are closed this implies that the lengths of all geodesics coincide. 
Urs Frauenfelder, Christian Lange and Stefan Suhr
math.DG math.SG