mp.mathematical physics - Is there a theory of differential equations for smooth correspondences?

This question is very closely related to another one I just asked. The general question is to what extent there is a theory of differential equations for smooth correspondences (between a smooth manifold and $mathbb R$, say). But I'm wondering about a specific well-studied case.

Background

Let $N$ be a compact smooth manifold with tangent bundle $TN$ and cotangent bundle $T^*N$. In the usual way, pick a Lagrangian $L: TN to mathbb R$, and suppose that it is nondegenerate in the sense that $frac{partial L}{partial v}$, thought of as a map $TN to T^*N$, is a fiber bundle isomorphism, where $v$ is a (vector of) fiber coordinate(s). Then, in the usual way, we can define a Hamiltonian $H: T^*N to mathbb R$ by $H = pv - L$, where $pv$ is the canonical pairing between a cotangent vector and a tangent vector, and I use $frac{partial L}{partial v}$ to identify $TN$ with $T^*N$. In the usual way, define on $N$ the second-order ordinary differential equation $frac{d}{dt}bigl[ frac{partial L}{partial v}bigr] = frac{partial L}{partial q}$ and $v = frac{dq}{dt}$, or equivalently $frac{dq}{dt} = frac{partial H}{partial p}$ and $frac{dp}{dt} = -frac{partial H}{partial q}$. (These are all written in local coordinates, where $q$ is a local coordinate on $N$ and $v = dq$ and $p = frac{partial}{partial q}$ are the corresponding coordinates on $TN$ and $T^*N$. But the ODE is coordinate-invariant.)

Since $N$ is compact and $L$ is nondegenerate, the ODE has global solutions, and each solution is determined by its initial conditions, which are given equivalently by a point in $TN$ and a point in $T^*N$. In the usual way, define an action map $S: TN times mathbb R to mathbb R$ by $S(v,q,t) = int_0^t L(phi(v,q,s))ds$, where $phi: TN times mathbb R to TN$ is the "flow" map for the ODE.

Let $pi$ be the projection $TN to N$, so that $pi(v,q) = q$, and using the flow map $phi$, define a map $TN times mathbb R$ to $N times N times mathbb R$ via $(v,q,t) mapsto (q,pi(phi(v,q,t)),t)$. Generically, this map is a local isomorphism, in the following sense: for generic $(v,q,t)$ (say, a dense open subset of $TNtimes mathbb R$ when $frac{partial^2 L}{partial v^2}$ is positive-definite), there is a small open neighborhood such that the map to $Ntimes N times mathbb R$ takes the neighborhood diffeomorphically to its image. Pick one such small neighborhood, and use it to push forward the action function $S$. Abusing notation, I will call this pushforward $S$. Then $S(q_1,q_2,t)$ satisfies the Hamilton-Jacobi equation: $frac{partial S}{partial t} = - H(frac{partial S}{partial q_2},q_2) = - H(-frac{partial S}{partial q_1},q_1)$.

My question

Above, I defined a local function $S: U to mathbb R$, where $U$ is a small neighborhood of $Ntimes Ntimes mathbb R$, and it satisfied a partial differential equation. But really I should have talked about the correspondence

$$ Ntimes Ntimes mathbb R overset{(pi, picirc phi, t)}{longleftarrow} TN times mathbb R overset{S}{longrightarrow} mathbb R $$

Is there language with which one can say that this correspondence satisfies the Hamilton-Jacobi equations? For example, what happens near non-generic (sometimes called "focal") points?

Bonus questions

(If $N$ is not compact, then the flow map does not have global-time solutions. But I can still do everything; I just have to replace the space $TN times mathbb R$ by an open subspace. In fact, $phi$ still defines on $TN times mathbb R$ the structure of an action groupoid, and both $(pi, picirc phi, t)$ and $S$ are groupoid homomorphisms, so that the above correspondence is a span of groupoids. Does this enter the discussion in any interesting way?)

If $frac{partial L}{partial v}$ does not define a bundle isomorphism, then the Hamiltonian is not well-defined as a function $H: T^*Nto mathbb R$. But it does make sense as a correspondence, by:

$$ T^*N overset{ frac{partial L}{partial v}}{longleftarrow} TN overset{ vfrac{partial L}{partial v} - L}{longrightarrow} mathbb R$$

Imposing the condition that $frac{partial^2 L}{partial v^2}(v,q)$ is an invertible matrix for each $(v,q)$, so that the ODE is still nondegenerate second-order, does the language of correspondences allow me to talk about the Hamilton-Jacobi equations when the Hamiltonian is not a function but the above correspondence?

Alternately, I could start with a function $H: T^*N to mathbb R$, and construct a correspondence $L$, etc. The Legendre transform should really be thought of as a transformation of correspondences, not of functions.

Finally, when $frac{partial^2 L}{partial v^2}$ is sometimes degenerate, then the ODE degenerates, sometimes in complicated ways. Can this be accommodated by making the flow $phi$ into a correspondence rather than a function?