I will first give a brief introduction of the connection between
a Hamilton-Jacobi equation and the Aubrey-Mather theory. This is the so
called weak KAM theory. An extremely interesting project in weak KAM
theory is to understand what kind of dynamical information is encoded in
the
effective Hamiltonian. I will present a result about the connection
between linear pieces on level curves of the effective Hamiltonian and the
structure of correspondent Aubry sets.