Maxwell's relations as Hamilton's equations: a symplectic and variational framework
Abstract
We develop a geometric framework in which classical thermodynamics is reformulated as a Hamiltonian dynamical system. An explicit canonical mapping (q,p,t,H)↔(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 ℒ, we prove that the closure condition dℒ=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.
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BibTeX
@misc{yookong2026maxwells,
title = {{Maxwell's relations as Hamilton's equations: a symplectic and variational framework}},
author = {Sikarin Yoo-Kong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18215},
eprint = {2609.18215},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18215},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maxwell's relations as Hamilton's equations: a symplectic and variational framework
AU - Sikarin Yoo-Kong
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2609.18215
UR - https://arxiv.org/abs/2609.18215
AB - We develop a geometric framework in which classical thermodynamics is reformulated as a Hamiltonian dynamical system. An explicit canonical mapping (q,p,t,H)↔(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 ℒ, we prove that the closure condition dℒ=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.
ER -