  In the present paper we compare the theory of mixtures based on Rational
Thermomechanics with the one obtained by Hamilton principle. We prove that the
two theories coincide in the adiabatic case when the action is constructed with
the intrinsic Lagrangian. In the complete thermodynamical case we show that we
have also coincidence in the case of low temperature when the second sound
phenomena arises for superfluid Helium and crystals.
