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$Q=\ln(p),\; P=p(1-q+\ln(p))$ was given.

I read that one needs to check whether the Poisson brackets $\{Q,P\}=1$ and $\{Q,Q\}=0=\{P,P\}$, but we haven't discussed that kind of method for approval. We did discuss the poisson bracket, and how a transformation is canonical, if a function $K(Q,P,t)$ exists which satisfies both equations:

$\dot{Q_i}=\frac{\partial K}{\partial p_i}$ and $\dot{P_i}=-\frac{\partial K}{\partial Q_i}$ , where $K$ is the Hamiltonian with the new generalised momenta $P$ and new generalised coordinates $Q$. Note that $p,q$ are the old momenta and coordinates. We have also discussed the 4 types of generating functions for a transformation, but I don't know how to solve the exercise with the methods presented in lecture. I don't know how to solve the exercise using these methods. I don't even know how to find out $K$ and all the more the generating function.

I would be very grateful if someone could help me somehow!

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  • $\begingroup$ Where do you read that one needs to check whether the Poisson bracket satisfies $\{Q,P\}=1$ and $\{Q,Q\}=0=\{P,P\}$? $\endgroup$
    – rreevv97
    Oct 18, 2021 at 5:21

1 Answer 1

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Hint: Your transformation may be written as $p=e^Q$, $P=e^Q (1-q+Q)$. This suggests one of the four types of generating functions. Use this to deduce the appropriate generating function and so verify that this is a canonical transformation.

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