Thermodynamic systems as extremal hypersurfaces

We apply variational principles in the context of geometrothermodynamics. The thermodynamic phase space TT and the space of equilibrium states EE turn out to be described by Riemannian metrics which are invariant with respect to Legendre transformations and satisfy the differential equations following from the variation of a Nambu–Goto-like action. This implies that the volume element of EE is an extremal and that EE and TT are related by an embedding harmonic map. We explore the physical meaning of geodesic curves in EE as describing quasi-static processes that connect different equilibrium states. We present a Legendre invariant metric which is flat (curved) in the case of an ideal (van der Waals) gas and satisfies Nambu–Goto equations. The method is used to derive some new solutions which could represent particular thermodynamic systems.