HAMILTON-JACOBI FORMALISM FOR THERMODYNAMICS: A DETAILED THEORETICAL AND APPLIED STUDY

Abstract

This study develops the Hamilton-Jacobi (HJ) formalism specifically for thermodynamics, inspired by its effectiveness in classical mechanics and optics. By recasting thermodynamic laws within a geometric context using contact manifolds and Legendre transformations, traditional equations of state emerge naturally as solutions to the HJ equation. Detailed examples for the ideal gas, van der Waals gas, and Curie-Weiss magnet demonstrate the method’s capability and insight. Discussions include the concept of ”thermodynamic time,” the benefits of a geometric unified viewpoint, and potential avenues for future research in generalized and quantum thermodynamics.