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Show that a transformation is canonical

by fluidistic
Tags: canonical, transformation
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fluidistic
#1
Feb29-12, 04:13 PM
PF Gold
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P: 3,188
1. The problem statement, all variables and given/known data
Show that the following transformation is canonical:
[itex]Q=\ln \left ( \frac{\sin p }{q} \right )[/itex], [itex]P=q \cot p[/itex].


2. Relevant equations
A transformation is canonical if [itex]\dot Q=\frac{\partial H'}{\partial P}[/itex] and [itex]\dot P =-\frac{\partial H'}{\partial Q}[/itex].
H' is the Hamiltonian in function of Q and P.

3. The attempt at a solution
I've calculated [itex]\dot Q = \dot p \cot p - \frac{\dot q \sin p }{q}[/itex] and [itex]\dot P=\dot q \cot p - \frac{q \dot p}{\sin ^2 p}[/itex].
Now I guess I must check out if the conditions about the partial derivatives of the Hamiltonian are satisfied but I do not know either H(q,p) nor do I know H'(Q,P). I'm stuck here.
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fluidistic
#2
Mar2-12, 03:57 PM
PF Gold
fluidistic's Avatar
P: 3,188
Ok guys, discard my attempt in my last post.
I've found in some class notes and in Landau&Lifgarbagez's book the necessary condition for a transformation to be canonical. If I'm not wrong, I must show that
I)[itex][Q,P]_{q,p}=1[/itex]
II)[itex][Q,Q]_{q,p}=0[/itex]
III)[itex][P,P]_{q,p}=0[/itex].
Where [itex][f,g]_{p,q}[/itex] denotes the Poisson brackets of f and g with respect to p and q. In other words, this is worth [itex]\frac{\partial f}{\partial p} \frac{\partial g}{\partial q}-\frac{\partial f}{\partial q} \frac{\partial g}{\partial p}[/itex].
Now I have that [itex]Q=\ln \left ( \frac{\sin p }{q} \right )[/itex] and [itex]P=q \cot p[/itex].
I tried to be careful in doing the partial derivatives. I found out that:
(1)[itex]\frac{\partial Q}{\partial p}=q \cot p[/itex]
(2)[itex]\frac{\partial P}{\partial q}= \cot p[/itex]
(3)[itex]\frac{\partial Q}{\partial q}=-\frac{1}{q}[/itex]
(4)[itex]\frac{\partial P}{\partial p}=q(-1- \cot ^2 p)[/itex]

Using this, I found out I) to be worth [itex](1-q) \cot ^2 p-1[/itex]. Unfortunately this isn't worth 1. What am I doing wrong?

Edit: I just found a mistake in my definition of Poisson's brackets. But now I find I) to be worth [itex]1+ \cot ^2 p-q \cot ^2 p[/itex] which is still wrong. I've rechecked the partial derivatives, I do not see any mistake, yet my result keeps being wrong.

Edit 2: This can be rewritten as [itex]\frac{1}{\sin ^2 p}-q \cot ^2 p[/itex]. This is not worth 1 so that the transformation isn't canonical, which is absurd. I do not see any mistake.

Edit 3 : I am nut guys! I found out the mistake in a derivative. I now reach all the results I should, problem solved!


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