I'm having some difficulty interpreting the mathematical underpinnings of this derivation given in chapter 4, section 16.
We have $F(x, y_1,...,y_n, y'_1,...,y'_n), p_i=F_{y'_i}$, and the familiar definition of $H:=-F+\sum_{i=1}^ny'_ip_i$. But in the next step, the following equation shows up:
$dH=-dF+\sum_{i=1}^np_idy'_i+\sum_{i=1}^ny'_idp_i$
This is said to be the "differential of the function $H$". Am I to understand this in the sense of a differential form (1-form)? In that case, what is the natural choice of manifold, and how should $F, y_i, y'_i$ be understood appropriately?
If $y_i$ are local coordinates on a manifold $X$, $y_i,y_i'$ are local coordinates on $TX$, in the sense that $\frac{\partial}{\partial y_i}|_{y}$ form a basis for the tangent space $T_yX$ at $y\in X$, and we take $\sum_{i} \frac{\partial}{\partial y_i}|_{y}y_i'\mapsto (y_i,y_i')$ to be a chart for $TX$. $F$ is a function on $TX$.