I'm trying to clarify how we get a Hamiltonian directly from a Lagrangian using the Legendre transform. Let me give some preliminaries for my question to make sense.
A Hamiltonian system is a triple $(M,\omega, H)$, where $(M,\omega)$ is a symplectic manifold and $H\in C^\infty(M)$ is some prescribed smooth function (the symplectic structure isn't important to this question, but since this is a symplectic topic, I'm going to include it).
Now, let $M$ be a smooth manifold and let $\omega$ be the standard symplectic form on $T^*M$, and let $F\in C^\infty(TM)$ be our Lagrangian, that is, $F_p:=F|_{T_pM}$ is strictly convex for each $p\in M$. For the moment, let's fix $p\in M$ and focus on $F_p:T_pM\to\mathbb{R}$. For $\alpha\in T_p^*M$, define $$F_{p,\alpha}(v)=F_p(v)-\alpha(v)$$ for $v\in T_pM$. The mapping $F_{p,\alpha}$ is said to be stable if there exists a unique critical point in $T_pM$ for $F_{p,\alpha}$. Then define the stability set $$S_{F_p}=\{\alpha\in T_p^*M:F_{p,\alpha}\text{ is stable}\}.$$ Then $S_{F_p}$ is open and convex in $T_p^*M$, and leads to my first question of when do we have that $S_{F_p}=T_p^*M$?
Furthermore, since $F_p$ is strictly convex, we have that the Legendre transform associated to $F_p$, denoted $L_{F_p}:T_pM\to S_{F_p}$ is a diffeomorphism. Moreover, with our dual $F_p^*:S_{F_p}\to\mathbb{R}$, we see that $$L_{F_p}^{-1}=L_{F_p^*}.$$ Again, with my understanding, don't we need $S_{F_p}=T_p^*M$ for this to make sense?
Combining the fibewise defined maps, we obtain $$\mathcal{L}:TM\to T^*M,\qquad \mathcal{L}|_{T_pM}=L_{F_p},$$ and $$H:S_F\to\mathbb{R},\qquad H|_{S_{F_p}}=F_p^*,$$ and $\mathcal{L}:TM\to S_F$ is a diffeomorphism.
Note that I'm using the notation of $$S_F=\bigcup_{p\in M}S_{F_p}.$$
Now, this defined $H$ should yield the Hamiltonian system $(T^*M,\omega,H)$, but we don't necessarily have $H$ defined on the whole cotangent bundle, which seems odds since our Lagrangian was defined on the whole tangent bundle. Am I missing something in this construction? I was under the impression that Lagrangians on the tangent bundle were equivalent to Hamiltonians on the cotangent bundle.
I guess all of this reduced to the question: How do we remedy the fact that $S_{F_p}$ and $T_p^*M$ aren't necessarily equal?
As always, any help or references would be greatly appreciated.