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The question I am asking was asked here but was never given a satisfactory answer and so I will rephrase it and add more detail.

In chapter 9 of Goldstein,when talking about a canonical transformatio $(p,q,t) \to (P,Q,t)$ where Hamilton's principle is obeyed:

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He mentions that (9.8) is a sufficient but not necessary condition.

However in chapter 10 between eqs. (10.1) and (10.2) he states that

enter image description here

which we derived using (9.8) with $\lambda=1$. How has the sufficient condition become a necessary one?

I though the idea of (9.8) is that if the integrands are both equal it is one way to satisfy (9.6) and (9.7) but not the only way?

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This is because Ref. 1. later simply declares in the text under eq. (9.11) that an canonical transformation (CT) by definition satisfies eq. (9.8) with $\lambda=1$.

For the various definitions of a canonical transformation (CT), see e.g. this Phys.SE post.

References:

  1. H. Goldstein, Classical Mechanics; Chapter 9. See text under eq. (9.11).
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