Hamiltonian structure of thermodynamics with gauge

The state of a thermodynamic system being characterized by its set of extensive variables q (i = 1, ..., n) , we write the associated intensive variables γi, the partial derivatives of the entropy S (q, ..., q) ≡ q0, in the form γi = −pi/p0 where p0 behaves as a gauge factor. When regarded as independent, the variables q, pi (i = 0, ..., n) define a space 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 M of T. Any thermodynamic process, even dissipative, taking place on M is represented by a Hamiltonian trajectory in T, governed by a Hamiltonian function which is zero on M. A mapping between the equations of state of different systems is likewise represented by a canonical transformation in T.Moreover a natural Riemannian metric exists for any physical system, with the q’s as contravariant variables, the pi’s as covariant ones. Illustrative examples are given.