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I’ve tried to show you that the symplectic structure on the phase spaces of classical mechanics, and the lesser-known but utterly analogous one on the phase spaces of thermodynamics, is a natural outgrowth of utterly trivial reflections on the process of minimizing or maximizing a function S on a manifold Q.
The first derivative test tells us to look for points with
$$d S = 0$$
while the commutativity of partial derivatives says that
$$d^2 S = 0$$
everywhere—and this gives Hamilton’s equations and the Maxwell relations.
https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/