
#1
Dec1706, 04:49 AM

P: 286

1. The problem statement, all variables and given/known data
Consider a canonical transformation with generating function [tex]F_2 (q,P) = qP + \epsilon G_2 (q,P)[/tex], where [tex]\epsilon[/tex] is a small parameter. Write down the explicit form of the transformation. Neglecting terms of order [tex]\epsilon^2[/tex] and higher,find a relation between this transformation and Hamilton's equations of motion, by setting [tex]G_2=H[/tex] (why is this allowed?) and [tex]\epsilon = dt[/tex]. 2. The attempt at a solution I think the transformation equations are [tex]\delta p = P  p = \epsilon \frac{\partial G_2}{\partial q}[/tex] and [tex]\delta q =Qq=\epsilon \frac{\partial G_2}{\partial q}[/tex] [tex]\dot{p}=\frac{\partial H}{\partial q}[/tex] and [tex]\dot{q}=\frac{\partial H}{\partial P}[/tex] which are the Hamiltonian equations of motion? And why is this allowed? 



#2
Dec1706, 08:21 AM

Emeritus
Sci Advisor
PF Gold
P: 6,238

Ooops !
I'm terribly sorry, instead of replying, I erroneously edited your post ! 



#3
Dec1706, 09:13 AM

P: 286

Yes it should be a P, not a q. I get the relations
[tex]Q = q + \delta q[/tex] and [tex]P = p + \delta p[/tex] but why is it allowed to use G = H? 



#4
Dec1706, 10:04 AM

Emeritus
Sci Advisor
PF Gold
P: 6,238

Canonical transformationThe reason for that is that your transformation equations you've found for an arbitrary G are what they are, and become the Hamilton equations of motion when you pick G to be equal to H. Now, strictly speaking we should write H(q,P) instead of H(q,p), but we can replace the P by p here, because they are only a small amount different. 



#5
Dec1806, 03:24 PM

P: 173

and it also preserve the fundamental poisson brackets....
its just a quick calcolus {Q,P} 


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