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A similar question has already been asked here: Derivation of Dirac-Delta with complicated argument $\delta(f(x))$

But now I want to do the following. Given the Hamiltonian $H=H(q(t),p(t),t)=E$ is a constant we can define its phase space volume as: $$\int \delta(H(q(t),p(t),t)-E)dqdp$$ How can one evaluate the rate of change of the phase space volume, considering that q and p are functions of time? $$\frac{d}{dt}\int \delta(H(q(t),p(t),t)-E)dqdp$$

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eeqesri
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  • What is the range of the integral? – pregunton Mar 24 '17 at 10:25
  • all of real space – eeqesri Mar 24 '17 at 14:51
  • Then, unless I'm missing something, I don't think the integral makes sense as stated. For me, $x=x(t), y=y(t)$ says that $(x,y)$ defines a unique curve parametrized by $t\in \mathbb{R}$, while $\int ... dx(t) dy(t)$ says that $(x,y)$ varies over all of $\mathbb{R}^2$. But both aren't possible at the same time. For example, what would be $\int e^{-(x(t)^2+y(t)^2)} dx(t) dy(t)$ if $x(t)=y(t)=t$?

    If $\int ... dx(t) dy(t)$ means integrating over all possible curves (like in a path integral) then the integral makes sense but the output doesn't depend on $t$ so the derivative is 0.

     – pregunton Mar 24 '17 at 16:00
  • let me edit the question so it makes more sense (please see above) – eeqesri Mar 24 '17 at 18:22
  • Take a look at the co-area formula. – Ian Mar 24 '17 at 18:37
  • thanks. my guess is that that will help. I have a question thought. which area of mathematics deals with these kind of manipulations? is it functional analysis? – eeqesri Mar 24 '17 at 18:41
  • Distribution theory and geometric measure theory. Functional analysis is related, though. – Ian Mar 24 '17 at 18:51

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