A Hamiltonian Formulation of Equilibrium Thermodynamics: From Gibbs Variational Principle to Analytical Thermodynamics

Analytical thermodynamics provides a study of thermodynamics through the methods of analytical mechanics. With it the canonical equations in equilibrium thermodynamics, i.e. the basic Poisson's brackets can be proved. The major objective is the discussion of canonical transformation of thermodynamical quantities in equilibrium thermodynamics by the Gibbs variational principle. These canonical transformations can take four forms. The quasi-Hamilton-Jacobi equation can be expressed and the canonical transformation technique, viz. that of changing a quasi-Hamiltonian into a pressure or a volume in equilibrium thermodynamics is performed. As an example of applying the canonical transformation, ideal gas is discussed giving clear and distinct results.