  We study the long time behavior of small solutions of semi-linear dispersive
Hamiltonian partial differential equations on confined domains. Provided that
the system enjoys a new non-resonance condition and a strong enough energy
estimate, we prove that its low super-actions are almost preserved for very
long times. Roughly speaking, it means that, to exchange energy, modes have to
oscillate at the same frequency. Contrary to the previous existing results, we
do not require the solutions to be especially smooth. They only have to live in
the energy space. We apply our result to nonlinear Klein-Gordon equations in
dimension d = 1 and nonlinear Schr{\"o}dinger equations in dimension d $\le$ 2.
