  We study the modulational instability of periodic traveling waves for a class
of Hamiltonian systems in one spatial dimension. We examine how the Jordan
block structure of the associated linearized operator bifurcates for small
values of the Floquet exponent to derive a criterion governing instability to
long wavelengths perturbations in terms of the kinetic and potential energies,
the momentum, the mass of the underlying wave, and their derivatives. The
dispersion operator of the equation is allowed to be nonlocal, for which Evans
function techniques may not be applicable. We illustrate the results by
discussing analytically and numerically equations of Korteweg-de Vries type.
