  We investigate the effects of Hamiltonian and Langevin microscopic dynamics
on the growth laws of domains in coarsening. Using a one-dimensional class of
generalized $\phi^4$ models with power-law decaying interactions, we show that
the two dynamics exhibit scaling regimes characterized by different scaling
laws for the coarsening dynamics. For Langevin dynamics, it concurs with the
exponent of defect dynamics, while Hamiltonian dynamics reveals new scaling
laws with distinct early-time and a late-time regimes. This new behaviour can
be understood as an effect of energy conservation, which induces a coupling
between the dynamics of the local temperature field and of the order parameter.
