he lecture is concerned with the study of periodic and
  subharmonic
solutions of some second order Hamiltonian systems
of the form
\[
-u''(t)+\nabla _uG(t,u)=0,
\]
where $G$ is periodic of period $T$ with respect to the time variable.
I shall discuss a critical point theorem for noncoercive functionals
which may be applied to obtain the existence of periodic solutions of
such systems. It will also be applied to the  special case of a one-dimensional
 nonlinear oscillator
\[
-u''(t)+g(t)|u|^{\alpha -2}u=0,~\alpha >1,
\] where $g$ is a periodic function  which assumes both
positive and negative values. In this case it will be shown that
subharmonic solutions exist of arbitrarily large periods.