Classical mechanics: Generating function of lagrangian submanifold I have a short question regarding the geometrical interpretation of the Hamilton-Jacobi-equation.
One has the geometric version of $H \circ dS = E$ as an lagrangian submanifold $L=im(dS)$, which is transverse to the fibers of $T^*Q$ and lies in the coisotropic submanifold $H^{-1}(E)$. Furthermore the pullback of the canonical one-form $\theta$ to L is exact.
The question is: Is L an embedded submanifold? I think if I have a solution S of the Hamilton-Jacobi-equation, then it generates an embedded submanifold $dS(Q)$. But have I have to use an embedded lagrangian submanifold to get an "analytical" solution or is an "normal" submanifold enough?
 A: Let $M$ and $N$ be manifolds and $f: M \to N$ a diffeomorphism. The Lagrangian submanifold generated by $f$ is 
$$ Y_f = \{(x,y,d_x f, d_y f); \, (x,y) \in M \times N\} \, ,$$
where
$$d_x f = (df)|_{T^*_x M} \quad \text{and} \quad d_y f = (df)|_{T^*_y N} \, . $$
Define the twist of $Y_f$ by
$$ Y_f^\sigma = \{(x,y,d_x f, -d_y f); \, (x,y) \in M \times N\} \, .$$
Theorem: $f$ is a generating function of the symplectomorphism $\phi: T^*M \to T^*N$ iff $Y_f^\sigma$ is its graph. This condition is equivalent to solving
$$ \phi(x,\xi) = (y,\eta) \Longleftrightarrow \xi = d_x f \quad \text{and} \quad \eta = -d_y f \, ,$$
which is a property of generating functions that can also be derived from the Hamilton-Jacobi (HJ) equation. 
This digression shows that not all Lagrangian submanifolds can be associated to generating functions. Since we are talking about differential manifolds and submanifolds that are graphs of diffeomorphisms, the topology of such submanifolds is inherited from the base.
In resume: the submanifold itself is not enough to guarantee that the HJ equation can be solved, but a solution to the HJ equation will definitely provide a well behaved Lagrangian submanifold.
