Question about canonical transformation and generating functions

Question

In Goldsteins' Mechanics, page 371 (relevant part appears below), it follows from what he states in the first yellow part that the equations of transformation:

$$Q = Q(q, p,t), \quad P = P(q, p,t)\tag{9.4}$$

are known, since he says that we can express $F$ as a function of either coordinate because we know eqs. (9.4) and its inverse relation.

In the second yellow part, he says that the generating function $F$ specifies the equation of transformation. But if as said in the first yellow part we know the transformation, why the generating function will give us something else (Why he says that the generating function specifies the eq. of tran.)? And if we say that the transformation equations result from the generating function, the first part has no meaning, since the transformation equations are not known until we find the relation between the two sets of canonical coordinates as follows from the derivative relations from the generating function.

Could someone explain what Goldstein means and clarify this?

Answer 1

1. Even if we assume that the canonical transformation (CT) (9.4) exists, it doesn't mean that we explicitly know its form.

2. Concerning OP's 1st yellow quote: Ref. 1 is not claiming that all types 1-4 of generating functions exist; only that some (or perhaps a hybrid thereof) exist locally, cf. e.g. this and this Phys.SE posts.

3. Example: The eq. (9.11) with the assumption that the generating function $$F(q,Q,p(q,Q,t),P(q,Q,t),t)~=~F_1(q,Q,t) \tag{9.12}$$ is of type 1 leads to conditions (9.14), which can help us solve for the explicit form of the CT (9.4), cf. OP's 2nd yellow quote. See also e.g. this related Phys.SE post.

References:

1. H. Goldstein, Classical Mechanics; section 9.1.

Answer 2

Perhaps this interpretation of Goldstein's intentions will be useful to you.

Halving the number of independent arguments

« in the first yellow part …he says that we can express F as a function of either coordinate because we know eqs. (9.4)»

Goldstein meant that if you first define some specific function $F$ of all possible four independent variables $F(q,p,Q,P)$, then from the previously known canonical transformations (9.4) it will be possible to express any pair of variables in terms of another pair variables, and thus leave only 2 independent variables (instead of 4).

Сombine half the old arguments and half the future new ones

«Why he says that the generating function specifies the eq. of tran.?»

Next, he is going to move in the opposite direction - to obtain canonical transformations (9.4) from various predefined $F$. He says that the combinations $F(q,Q), F(q,P), F(p,Q), F(p,P)$ turn out to be especially useful for this (later they were called generating functions $F_1$, $F_2$, $F_3$ and $F_4$). In the second part, highlighted in yellow, he proceeds to obtain the missing pair of canonical variables (and all relations of type 9.4) from a known function of exactly this useful type (from two arguments, not four) using the example of $F_1(q,Q)$. Thus, the phrase that confused you about substituting the known relations (9.4) into the function $F$ was needed by Goldstein to illustrate 1) the usefulness of halving the number of independent arguments of the functions $F$ and 2) the usefulness of introducing such functions (generating functions $F_1-F_4$), in which half arguments are taken from old variables, and half from new ones, because they serve as the most convenient bridges for transition between all old and all new canonical variables.