Equilibrium Thermodynamics, Maxwell relations and Legendre submanifolds

In this survey we study equilibrium thermodynamics through the lens of contact geometry. On the Gibbs space with contact 1-form [Formula: see text], equilibrium states constitute a 2-dimensional Legendre submanifold [Formula: see text] characterized by [Formula: see text]. In particular, in this artcle we study the connection between the Legendre submanifolds associated to different thermodynamic potentials and Maxwell relations. Using exterior calculus, we show that Maxwell relations are precisely the integrability conditions [Formula: see text]. and the pullback of two form vanishes on the Legendre submanifold [Formula: see text]. We describe Weinhold or Ruppeiner metric on the thermodynamic phase space associated to thermodynamic potentials. We also give a brief account of Callen-Tisza treatment. We explore contact Hamiltonian mechanics and symplectic mechanics using the map between contactomorphism and symplectomorphism. The symplectization of thermodynamics is given via the map between contact form and Liouville form. This unifies Maxwell integrability, Legendre invariance of potentials, and metric stability within a single contact–symplectic framework.