IN [l], Ekeland and Hofer stated a very general result for the existence of periodic solutions with any prescribed minimal period, for Hamiltonian systems, assuming that the Hamiltonian function H is strictly convex, with superquadratic behaviour at the origin and at infinity. The arguments given in [l] are based on an estimate for the Morse index of a critical point of Mountain Pass type for a functional which is associated with the Hamiltonian system via the duality principle of Clarke and Ekeland [2]. Unfortunately, the functional is only C’ and not C”, then the result is achieved only through a finite dimensional reduction and, as the authors themselves say, “through an inordinate amount of technicalities”. In this paper, we give a “direct” definition of the Morse index of a periodic solution which does not use the duality principle. Indeed, it is based on a study of the Morse indexes of approximating solutions obtained by a Galerkin scheme such as the one introduced by Rabinowitz in [3]. Taking into account some results about linking point theory recently stated in [4-71, we get the same index estimates as in [l], but in a simple way. Moreover, some generalizations are obtained, in the sense that the convexity assumption on H can be replaced either by a local convexity condition or by a condition on the second derivative of H, plus, eventually, a symmetry condition. (See also [8, 91 for some other results obtained in this framework.) In particular, in the locally convex case, the index estimates yield direct information on the minimal period of the corresponding solution. Finally, let us remark that this definition of the Morse index can be adapted to other nonconvex cases, and one can still get some index estimates of the same type.