# How to check whether this transformation is canonical?

$$Q=\ln(p),\; P=p(1-q+\ln(p))$$ 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!

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

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.