  In this paper, we construct two classes of planar polynomial Hamiltonian
systems having a center at the origin, and obtain the lower bounds for the
number of critical periods for these systems. For polynomial potential systems
of degree $n$, we provide a lower bound of $n-2$ for the number of critical
periods, and for polynomial systems of degree $n$, we acquire a lower bound of
$n^2/2+n-5/2$ when $n$ is odd and $n^2/2-2$ when $n$ is even for the number of
critical periods. To the best of our knowledge, these lower bounds are new,
moreover the latter one is twice the existing results up to the dominant term.
