Keplerian orbits through the Conley-Zehnder index

It was discovered by Gordon (Am J Math 99(5):961–971, 1977) that Keplerian ellipses in the plane are minimizers of the Lagrangian action and spectrally stable as periodic points of the associated Hamiltonian flow. The aim of this note is to give a direct proof of these results already proved by authors in Hu and Sun (Adv Math 223(1):98–119, 2010), Hu et al. (Arch Ration Mech Anal 213(3):993–1045, 2014) through a self- contained and explicit computation of the Conley–Zehnder index through crossing forms in the Lagrangian setting. The techniques developed in this paper can be used to investigate the higher dimensional case of Keplerian ellipses, where the classical variational proof no longer applies.