I have a volume element in phase space:

$$ d\omega = \prod _{i=1}^{N}(dq_{i},dp_{i})$$

Now I should show the invariance of this product under canonical transformations. I think first I would have to write down a general equation for the canonical transformation then compute the total differential and plug this into the equation given. My problem is now I can't figure out how the equation for a general canonical transformation would look like. Does anybody have a hint how to start? The proof should be done without the concept of Symplectomorphisms like it was already shown in this question Which transformations are canonical?

  • $\begingroup$ This question (v2) is a duplicate of the second subquestion in this Phys.SE post. $\endgroup$ – Qmechanic May 11 '18 at 6:50
  • $\begingroup$ @Qmechanic I read this question but it does not really answer the question to me since I am trying to find this property without the concept of Symplectomorphisms. $\endgroup$ – zodiac May 11 '18 at 7:08
  • $\begingroup$ So which definition of CT do you use then? $\endgroup$ – Qmechanic May 11 '18 at 7:49
  • $\begingroup$ I was thinking to define the CT as a transformation that preserves the poisson bracket. $\endgroup$ – zodiac May 11 '18 at 8:45
  • 3
    $\begingroup$ That is the defining property of a symplectomorphism. $\endgroup$ – Qmechanic May 11 '18 at 22:15

Consider a phase space volume $d\omega = \prod dq^idp_i$. Time evolution can be viewed as the unfolding of canonical transformations, hence, we shall evolve this system $t\rightarrow t+dt$. We wish to show invariance of the measure under this canonical transformation $(q,p) \rightarrow (Q,P)$ where: $$ Q^i = q^i + \dot q^id t, \qquad P^i = p_i + \dot p_idt $$ Compute the new volume $d\omega' = \prod dQ^idP_i$:

\begin{align} d\omega ' &= \prod d( q^i + \dot q^id t, p_i + \dot p_id t)\\ & = \prod ( dq^i + \frac{\partial \dot q^i}{\partial q^i}dt, dp_i + \frac{\partial \dot p_i}{\partial p_i}d t) \end{align} Let us now expand this and retain terms linear in $dt$ we obtain $$ d\omega' \approx \sum_i \bigg( \frac{\partial \dot q^i}{\partial q^i} + \frac{\partial \dot p_i}{\partial p_i}\bigg) = \bigg\{\frac{\partial }{\partial q^i}\frac{\partial H}{\partial p_i} - \frac{\partial }{\partial p_i}\frac{\partial H}{\partial q^i}\bigg\} = 0 $$ Therefore, we have shown invariance of the measure under a canonical transformation.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.