I have recently learned about holonomic and nonholonomic constraints in analytical mechanics, and how they can be expressed as exact (Pfaffian form); for example:
$${\displaystyle df_{i}=\sum _{j}\ A_{ij}\,du_{j}+A_{i}\,dt=0}\tag{1}$$
and non-exact differentials, respectively.
I also know about state and path variables in thermodynamics, and how they can be expressed as exact or non-exact differentials, respectively. I have noticed a similarity between these two concepts.
My conclusion is that holonomic equations and nonholonomic equations are what the state variables and path variables are in thermodynamics, respectively.
The differential form $dU$ is a state function, i.e. you can integrate it completely and it is a closed form and a Pfaffian differential form. On the other hand $\delta Q$ and $\delta W$ are not completely integrable, i.e. they depend on the path taken to achieve them, exactly as nonholonomic systems depend on the path taken to achieve them. They are based on contact geometry.
My question: "Is there a deeper connection behind this observation?",
or is this just a coincidence behind the fact that both physical theories use differential forms and similar concepts in their own formulation?
Appendix:
are the Universal test for holonomic constraints and the Test for exactness of differential forms using the Schwarz symmetry of second derivatives also connected?
Because this:
$$\begin{equation*} \sum_{i=1}^k \frac{\partial f_i}{\partial x_j} \delta x_j + \frac{\partial f_i}{\partial y_j} \delta y_j + \frac{\partial f_i}{\partial z_j} \delta z_j = 0 \end{equation*}$$
reminds me of this
$$\frac{∂M}{∂y} = \frac{∂N}{∂x}.$$
