Why are the integrability conditions necessary and sufficient for the existence of a canonical transformation's generating function?

1. Let us for later convenience introduce a collective notation $$ (Z^1,ldots,Z^{2n}) ~=~ (Q^1, ldots, Q^n;P_1,ldots, P_n) $$ for the new phase space variables.

2. Next note that the integrability conditions (24a-c) are Maxwell relations $$tag{24'} frac{partial Theta_I}{partial Z^J} ~=~(I leftrightarrow J), qquad I,J~in~{1,ldots, 2n}, $$ for some functions $Theta_I=Theta_I(Z,t)$, whose explicit form is given in Ref. 1.

3. Equivalently, the one-form $$ Theta~=~sum_{I=1}^{2n} Theta_I ~mathrm{d}Z^I $$ is closed $$ tag{24''} mathrm{d}Theta~=~ 0. $$

4. Poincare Lemma then states that there exists$^1$ a function/zero-form $F=F(Z,t)$ such that $$ tag{23''}Theta~=~mathrm{d}F. $$

5. Equivalently, $$ tag{23'} Theta_I~=~frac{partial F}{partial Z^I}, qquad I~in~{1,ldots, 2n}, $$ which is the content of eqs. (23a-b) in Ref. 1.

6. Conversely, if $F$ exists, then $Theta$ is closed, since $mathrm{d}^2=0$. Hence the integrability conditions (24a-c) are necessary and sufficient$^1$ condition for the existence of $F$.

References:

1. E.C.G. Sudarshan & N. Mukunda, Classical Dynamics: A Modern Perspective, 1974; p.36.

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$^1$ Mathematical caveat: The existence of the function $F$ is only guaranteed in a contractible region of phase space.