Maxwell's relations as Hamilton's equations: a symplectic and variational framework
We develop a geometric framework in which classical thermodynamics is reformulated as a Hamiltonian dynamical system. An explicit canonical mapping $(q,p,t,H)\leftrightarrow(V,-P,S,-T)$ identifies Maxwell's relations as the characteristic equations of the thermodynamic Poincaré--Cartan one-form, in direct parallel with Hamilton's equations of motion. Treating thermodynamic potentials as action functionals, a variational principle recovers both the Maxwell relations and the thermodynamic constraints (adiabaticity, isothermality) as conserved first integrals. For adiabatic processes in an ideal gas, this yields explicit second-order ordinary differential equations for $V(T)$ and $P(T)$, each governed by a temperature-dependent Lagrangian. Assembling the component Lagrangians into a multi-parameter Lagrangian 1-form $\mathcal{L}$, we prove that the closure condition $d\mathcal{L}=0$ holds on the solution manifold, establishing that path-independence of thermodynamic state functions is a geometric consequence of multi-time integrability rather than an independent postulate.