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Logarythmic
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Homework Statement
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 =Q-q=\epsilon \frac{\partial G_2}{\partial q}[/tex]
vanesch said:I guess there's a typo here:
[tex]\delta q =Q-q=\epsilon \frac{\partial G_2}{\partial P}[/tex]
but how can I solve the last part? Can I just say that with the use of H and dt the equations can be written as
[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?
vanesch said:The idea is that we work in first order in [tex]\epsilon[/tex], and that you can hence replace everywhere [tex]P[/tex] by [tex]p[/tex] as the difference will introduce only second-order errrors.
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