There are a couple folks at work who are also interested in geometry and thermo. Their motivation is two fold. On the one hand they’d like to understand thermo better (thermo potentials especially) and also to develop methods to help students understand thermo better.
Increasingly I think what this will require is for us (the interested faculty/grad students/and I) to meet regularly to discuss the more advanced attempts to ‘geometrize’ thermo and then once we understand it, attempt to boil it down to something that will be pedagodgically useful.
I think this is possible, consider the last course in calculus that students take before moving on to differential equations. Its essentially a differential geometry course stripped down and made calculation intensive. So can we take the language of manifolds, bundles, etc for thermo and boil it down to something calculus like for the students?
After a recent email exchange with a faculty member I have started to wonder how we should think about the thermo potentials. Should we be starting from the idea that U (internal energy) is a function on the manifold or that dU is an exact differential form on the manifold (First Law).
Also, consider the case in mechanics where the Lagrangian lives on the tangent bundle but the Hamiltonian lives on the cotangent bundle (phase space). The Legendre transformation connects these two via the fiber derivative of the Lagrangian. Is it possible that some of our thermo potentials live in different spaces? That is, does dU live in cotangent land but perhaps dH or dG or dA live elsewhere?