About the lift of irreversible thermodynamic systems to the Thermodynamic Phase Space

In this paper, we analyze the different lift of the dynamics of irreversible thermodynamic systems from the Legendre submanifold associated with the thermodynamic properties of the physical system to the full Thermodynamic Phase Space. Firstly, we define a set of Hamiltonian functions which generate a class of equivalent lifts, that it contact vector fields which are equal on some distinguished Legendre submanifold. Secondly we show that how one may construct an equivalent contact vector field which renders attractive of the inverse image of zero by the Hamiltonian function.