$\sin$ with angle subscript In reading Haar's book on the old quantum mechanics, I came across a derivation of the square of the (classical) angular momentum that I am having some trouble understanding.
He claims that the square of the angular momentum $M^2$ is given by
$M^2 = [\vec{r}\times m\dot{\vec{r}}]\cdot[\vec{r}\times m\dot{\vec{r}}] = p_\theta^2 + \frac{p_\theta^2}{sin^2_\phi\theta}$
where $r$, $\theta$ and $\phi$ are the usual spherical coordinates and $p_\theta=mr^2\dot{\theta}$
I have not ever encountered the notation $\sin_\phi\theta$ before and am unsure exactly what it should mean. There is no other reference to this in the book. Is there a standard definition for $\sin_\phi\theta$ for given angles of $\phi$ and $\theta$?
 A: $$ M^2= p_\theta^2+ p_\phi^2 /\sin^2\!\theta ,$$
where $p_\phi =mr^2\! \sin^2\! \theta ~~\dot{\phi}$, as well.
Looks like you have two, not one, typos in the second term.
A: you can calculate it yourself :
with sphere position vector
I)
$$\vec R=r\, \left[ \begin {array}{c} \cos \left( \varphi  \right) \sin \left( 
\vartheta  \right) \\ \sin \left( \varphi  \right) 
\sin \left( \vartheta  \right) \\ \cos \left( 
\vartheta  \right) \end {array} \right] 
$$
$$\vec{\dot R}=\frac{\partial  R}{\partial \vec q}\,\vec{\dot q}$$
$$\vec q= \left[ \begin {array}{c} \vartheta \\ \varphi 
\end {array} \right] 
~,\vec{\dot q}= \left[ \begin {array}{c} \dot\vartheta \\ \dot\varphi 
\end {array} \right] 
$$
$$\vec M=\vec R\times\,m\,\vec{\dot R}$$
$\Rightarrow$
$$M^2=\vec M\,\cdot\,\vec M={m}^{2}{r}^{4}{\dot\vartheta }^{2}+{m}^{2}{r}^{4}{\dot\varphi }^{2} \left( 
\sin \left( \vartheta  \right)  \right) ^{2}\overset{!}{=}
p_\vartheta^2+\frac{p_\varphi^2}{\sin^2(\vartheta)}$$
II)
interchange $\vartheta\Leftrightarrow\varphi$
$$\vec R=r\,\left[ \begin {array}{c} \cos \left( \vartheta  \right) \sin \left( 
\varphi  \right) \\ \sin \left( \vartheta  \right) 
\sin \left( \varphi  \right) \\ \cos \left( \varphi 
 \right) \end {array} \right] 
$$
$\Rightarrow$
$$M^2={m}^{2}{r}^{4}{\dot\vartheta }^{2} \left( \sin \left( \varphi  \right) 
 \right) ^{2}+{m}^{2}{r}^{4}{\dot\varphi }^{2}
\overset{!}{=}p_\varphi^2+\frac{p_\vartheta^2}{\sin^2(\varphi)}$$
