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In Chapter 9 of Arnold's Mathematical Methods of Classical Mechanics, we find the following definition:

Let $g$ be a differentiable mapping of the phase space $\mathbb R^{2n}$ to $\mathbb R^{2n}$. The mapping $g$ is called canonical if $g$ preserves the $2$-form $\omega^2 = \sum dp_i \wedge dq_i$.

My question is, what does it mean to preserve a differential form? If we write the mapping as $g : (p, q) \mapsto (P, Q)$, does it mean that $\sum dP_i \wedge dQ_i = \sum dp_i \wedge dq_i$?

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Yes, it does.

It's a narrow class of possible variable changes.

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  • $\begingroup$ Thanks. So for $n=1$, we would need $\frac{\partial P}{\partial p} \frac{\partial Q}{\partial q} - \frac{\partial P}{\partial q} \frac{\partial Q}{\partial p} = 1$? $\endgroup$
    – Alan C
    Commented Dec 25, 2012 at 22:32

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