No answers yet? So let's take a shot at a (partial) answer:
Therefore as my main question, does symplectic geometry underpin thermodynamics?
No. In thermodynamics, we're dealing with a Legendrian submanifold of a contact manifold (cf Wikipedia). Thermodynamic variables are canonical coordinates on that manifold.
Morally speaking, in case of symplectic geometry, canonical coordinates map any symplectic manifold to the cotangent bundle $T^*\mathbb R^n$ with symplectic form $\omega = d\theta$.
In case of contact geometry, canonical coordinates map any contact manifold to the first jet bundle $J^1\mathbb R^n$ (essentially $\mathbb R\times T^*\mathbb R^n$) with contact form $\alpha = dz + \theta$ (in both cases, $\theta$ denotes the canonical 1-form of the cotangent bundle; $z$ is the coordinate of the first factor).
On the jet bundle, the submanifold in question is given by the prolongation of some state function - a thermodynamic potential expressed in its natural variables. Eg for $U = f(S, V)$, we end up with a coordinate expression
$$
(S, V, U, T, p) = \left(S, V, f(S, V), \frac{\partial f}{\partial S}(S, V), \frac{\partial f}{\partial V}(S, V) \right)
$$
Please insert minus signs as appropriate ;)
[I] was wondering can indeterminism in perturbation theory and chaos lead to entropy and the second law?
As far as geometry is concerned, there isn't really anything special about entropy, ie this question has to be answered at the lower level of statistical mechanics; I'm happy to leave that part of the question to someone else...
