Dirac Distribution in Polar Coordinates as a Measure I have a question.
I was browsing different scripts on statistical mechanics and saw a definition of the "area of accessible phase space at energy $E$ " for a single harmonic oscillator. The author quickly corrected himself, stating, that this is more of a contour in phase space. So I calculated both, got different results and need someone explaining, where my mistake is or what is causing this discrepancy.
For simplicity let's define the Hamiltonian as:
$$
H(q,p) = p^2 + q^2
$$
And the curve $\mathcal{C}$ in phase space $\Gamma=\{(q,p)\in\mathbb{R}^2\}$ is the circle, where $H=E$ with the radius $R=\sqrt{E}$.
Now the odd defintion for the mentioned area / contour given by the mentioned textbook author, where $\delta$ is the Dirac Distribution:
$$
N(E)
=\int\mathrm{d}\Gamma~\delta(H(q,p)-E)
$$
If I remember correctly there is also a tool given by the Dirac $\delta$ to evaluate compositions with a function $f$ which will come in handy:
$$
\delta(f(x))
=\sum\limits_{i=1}^N\frac{\delta(x-x_i)}{|{f'(x_i)}|}
$$
Here $x_i$ denote the roots of $f(x)$.
My calculation would now read as follows:
\begin{align*}
N(E)
&=
\int\limits_{-\infty}^\infty
\mathrm{d}q
\int\limits_{-\infty}^\infty
\mathrm{d}p
~
\delta(p^2+q^2-R^2)
=
\int\limits_{0}^\infty
\mathrm{d}\rho
~
\rho
~
\delta(\rho^2-R^2)
\int\limits_{0}^{2\pi}
\mathrm{d}\varphi \\
&=
2\pi
\frac{1}{2R}
\int\limits_{0}^\infty
\mathrm{d}\rho
~
\rho
~
\delta(\rho-R)
=\pi
\end{align*}
Here I used the know parameterization $(q,p)^T=(\rho\cos\varphi,\rho\sin\varphi)^T$.
It's significant, that I get (if I'm not wrong) the area of a unit circle independent of the energy $E$.
That's completely against my intuition so I try it again with an line integral over $\mathcal{C}=\{(q,p)\in\mathbb{R}^2|q^2+p^2=R^2\}$.
$$
N(E)
=
\int\limits_\mathcal{C}
\mathrm{d}s
=
\int\limits_0^{2\pi}
\mathrm{d}\varphi
\left|
\frac{\partial \vec{s}}{\partial \varphi}
\right|
=
\int\limits_0^{2\pi}
\mathrm{d}\varphi
\left|
\begin{pmatrix}
-R\sin\varphi
\\
R\cos\varphi
\end{pmatrix}
\right|
=
2\pi R
=2\pi \sqrt{E}
$$
This is of course a very simple calculation in this case of a circle.
So now my questions: These two results are different right? If yes, why thinks the author of this textbook, he could define a contour in phase space with the help of this Dirac Distribution and if he didn't intend to do that, what is the use of an "area" defined like this (with a DD)?
 A: As I know little about the physical context that leads to the quantity $N(E)$, let me focus on the mathematical aspect. Note that
$$ \int_{\mathbb{R}^2} \mathrm{d}q\mathrm{d}p \, \delta(q^2+p^2-R^2)
= \left[ \frac{\partial}{\partial \epsilon} \int_{\mathbb{R}^2} \mathrm{d}q\mathrm{d}p \, \mathbf{1}_{\{ q^2 + p^2 - R^2 \leq \epsilon \}}  \right]_{\epsilon = 0}, $$
where $\mathbf{1}_{\{\cdots\}}$ is the indicator function notation, which takes value $1$ if $\cdots$ is true and value $0$ otherwise. On the other hand, the line integral in your second formulation corresponds to
$$ \int_{\mathcal{C}} \mathrm{d}s
= \int_{\mathbb{R}^2} \mathrm{d}q\mathrm{d}p \, \delta(\sqrt{\smash[b]{q^2+p^2}}-R)
= \left[ \frac{\partial}{\partial \epsilon} \int_{\mathbb{R}^2} \mathrm{d}q\mathrm{d}p \, \mathbf{1}_{\{ \sqrt{\smash[b]{q^2 + p^2}} - R \leq \epsilon \}}  \right]_{\epsilon = 0}. $$
In both cases, the integral is supported on the contour $H(q, p) = E$, but the surface measure is weighted in a different way, i.e., $ \mathrm{d}q\mathrm{d}p \, \delta(q^2+p^2-R^2) = \mathrm{d}q\mathrm{d}p \, \frac{1}{2R} \delta(\sqrt{\smash[b]{q^2 + p^2}}-R) $. Since this "energy shell" as a hypersurface in the phase space does not alone encode the information about the surface measure, it should be specified by other means. I guess that the definition
$$ N(E)
= \int \mathrm{d}\Gamma \, \delta(H(\Gamma) - E)
= \left[ \frac{\partial}{\partial \epsilon} \int \mathrm{d}\Gamma \, \mathbf{1}_{\{ H(\Gamma) \leq E + \epsilon \}}  \right]_{\epsilon = 0} $$
exactly serves this purpose, and perhaps the author's notion of "area" conforms to this choice.
