Hamiltonian structure of thermodynamics with gauge

The state of a thermodynamic system being characterized by its set of extensive variables $q^i(i=1,\mathrm{.}\mathrm{.}\mathrm{.},n),$ we write the associated intensive variables ${\gamma }_i,$ the partial derivatives of the entropy $S(q^1,\mathrm{.}\mathrm{.}\mathrm{.},q^n)\equiv q_0,$ in the form ${\gamma }_i=-p_i\mathrm{/}{p}_0$ where $p_0$ behaves as a gauge factor. When regarded as independent, the variables $q^i,p_i(i=0,\mathrm{.}\mathrm{.}\mathrm{.},n)$ define a space $\mathbb{T}$ having a canonical symplectic structure where they appear as conjugate. A thermodynamic system is represented by a $n+1$-dimensional gauge-invariant Lagrangian submanifold $\mathbb{M}$ of $\mathbb{T}\mathrm{.}$ Any thermodynamic process, even dissipative, taking place on $\mathbb{M}$ is represented by a Hamiltonian trajectory in $\mathbb{T},$ governed by a Hamiltonian function which is zero on $\mathbb{M}\mathrm{.}$ A mapping between the equations of state of different systems is likewise represented by a canonical transformation in $\mathbb{T}\mathrm{.}$ Moreover a natural Riemannian metric exists for any physical system, with the $q^i$'s as contravariant variables, the $p_i$'s as covariant ones. Illustrative examples are given.