classical mechanics - Independent canonical coordinate variables?

In Goldstein's Classical Mechanics (2nd ed.) on section 9-1 page 382, there is a discussion about finding a canonical transformation $(q_i,p_i)\rightarrow (Q_j(q_i,p_i,t),P_j(q_i,p_i,t))$ from a given generating function $F=F_1(q_i,Q_i,t)$. The following is written on the page:

Equation (9-11) then takes the form,

$$\begin{align*} p_i\dot{q}_i-H&=P_i\dot{Q}_i-K+\frac{dF_1}{dt},\\ &=P_i\dot{Q}_i-K+\frac{\partial F_1}{\partial t}+ \frac{\partial F_1}{\partial q_i}\dot{q}_i+\frac{\partial F_1}{\partial Q_i}\dot{Q}_i \tag{9-13} \end{align*}$$

Since the old and the new coordinates, $q_i$ and $Q_i$ , are separately independent, Eq. (9-13) can hold identically only if the coefficients of $\dot{q}_i$ and $\dot{Q}_i$ each vanish:

I don't understand the bold text above. How are $q_i$ and $Q_i$ separately independent? I thought $Q_i$ was, in general, a function of the original canonical coordinate variables $q_i$, thereby making it explicitly dependent on $q_i$ (not independent). Could somebody explain how these two canonical coordinate variables are separately independent?

Answer

Let us suppress $t$-dependence in this answer for simplicity. A canonical transformation$^1$ $$(q,p)\quad\longrightarrow\quad (Q(q,p),P(q,p))$$ can be viewed as a graph, thereby yielding a $2n$-dimensional submanifold $M$ embedded inside a $4n$-dimensional total manifold $N$. The total manifold $N$ has $4n$ local coordinates $(q,p,Q,P)$. The submanifold $M$ can in certain cases be viewed as a graph of a function $$(q,Q)\quad\longrightarrow \quad(p(q,Q),P(q,Q)),$$ where $q$ and $Q$ are independent variables, and where the functions $$p(q,Q)~=~\frac{\partial F_1(q,Q)}{\partial q} , \tag{9-14a}$$ $$P(q,Q)~=~-\frac{\partial F_1(q,Q)}{\partial P} , \tag{9-14b}$$ comes from an $F_1$-type generating function $F_1(q,Q)$.

References:

1. H. Goldstein, Classical Mechanics.

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$^1$ Whatever that means.