Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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 $\sum dP_i \wedge dQ_i = \sum dp_i \wedge dq_i$?

share|improve this question
add comment

1 Answer 1

Yes, it does.

It's a narrow class of possible variable changes.

share|improve this answer
    
Thanks. So for $n=1$, we would need $\frac{\partial P}{\partial p} \frac{\partial Q}{\partial q} - \frac{\partial P}{\partial q} \frac{\partial Q}{\partial p} = 1$? –  Alan C Dec 25 '12 at 22:32
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.