Editing Spherical coordinate system (section)

== Kinematics ==
In spherical coordinates, the position of a point or particle (although better written as a [[tuple|triple]]<math>(r,\theta, \varphi)</math>) can be written as<ref name="Cameron2019">{{Cite book |last=Reed |first=Bruce Cameron |url=https://www.worldcat.org/oclc/1104053368 |title=Keplerian ellipses : the physics of the gravitational two-body problem |date=2019 |others=Morgan & Claypool Publishers, Institute of Physics |isbn=978-1-64327-470-6 |location=San Rafael [California] (40 Oak Drive, San Rafael, CA, 94903, US) |oclc=1104053368}}</ref>
<math display="block">\mathbf{r} = r \mathbf{\hat r} .</math>
Its velocity is then<ref name="Cameron2019" />
<math display="block">\mathbf{v} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = \dot{r} \mathbf{\hat r} + r\,\dot\theta\,\hat{\boldsymbol\theta } + r\,\dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\varphi}}</math>
and its acceleration is<ref name="Cameron2019" />
<math display="block">
\begin{align}
\mathbf{a} = {} & \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} \\[1ex]
= {} & \hphantom{+}\; \left( \ddot{r} - r\,\dot\theta^2 - r\,\dot\varphi^2\sin^2\theta \right)\mathbf{\hat r} \\
& {} + \left( r\,\ddot\theta + 2\dot{r}\,\dot\theta - r\,\dot\varphi^2\sin\theta\cos\theta \right) \hat{\boldsymbol\theta } \\
& {} + \left( r\ddot\varphi\,\sin\theta + 2\dot{r}\,\dot\varphi\,\sin\theta + 2 r\,\dot\theta\,\dot\varphi\,\cos\theta \right) \hat{\boldsymbol\varphi}
\end{align}
</math>

The [[Angular_momentum#Orbital_angular_momentum_in_three_dimensions| angular momentum]] is 
<math display="block">  \mathbf{L} =
\mathbf{r} \times \mathbf{p} = \mathbf{r} \times m\mathbf{v} = m r^2 \left(- \dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\theta}} + \dot\theta\,\hat{\boldsymbol\varphi }\right) 
</math>
Where <math>m</math> is mass. In the case of a constant {{mvar|φ}} or else {{math|''θ'' {{=}} {{sfrac|{{pi}}|2}}}}, this reduces to [[Polar coordinate system#Vector calculus|vector calculus in polar coordinates]].

The corresponding [[Angular_momentum_operator#Orbital_angular_momentum_in_spherical_coordinates| angular momentum operator]] then follows from the phase-space reformulation of the above,
<math display="block"> \mathbf{L}= -i\hbar ~\mathbf{r} \times \nabla =i \hbar \left(\frac{\hat{\boldsymbol{\theta}}}{\sin(\theta)} \frac{\partial}{\partial\phi} - \hat{\boldsymbol{\phi}} \frac{\partial}{\partial\theta}\right). </math>

The torque is given as<ref name="Cameron2019" />
<math display="block">  \mathbf{\tau} =
\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \mathbf{r} \times \mathbf{F} = -m \left(2r\dot{r}\dot{\varphi}\sin\theta + r^2\ddot{\varphi}\sin{\theta} + 2r^2\dot{\theta}\dot{\varphi}\cos{\theta} \right)\hat{\boldsymbol\theta} + m \left(r^2\ddot{\theta} + 2r\dot{r}\dot{\theta} - r^2\dot{\varphi}^2\sin\theta\cos\theta \right) \hat{\boldsymbol\varphi}
</math>

The kinetic energy is given as<ref name="Cameron2019" />
<math display="block"> E_k =
\frac{1}{2}m \left[ \left(\dot{r}\right)^2 + \left(r\dot{\theta}\right)^2 + \left(r\dot{\varphi}\sin\theta\right)^2 \right]
</math>