Accelerator Physics/Coordinate Systems

Cartesian coordinates
$$\mathrm{d}V = \mathrm{d}x\mathrm{d}y\mathrm{d}z$$

$$\vec{\nabla}\psi = \left(\frac{\partial\psi}{\partial x}, \frac{\partial \psi}{\partial y}, \frac{\partial \psi}{\partial z}\right) $$

$$\vec{\nabla}\cdot\vec{v} = \frac{\partial v_x}{\partial x}+ \frac{\partial v_y}{\partial y}+ \frac{\partial v_z}{\partial z} $$, sometimes denoted as $${\nabla}\mathrm{v} $$

$$\vec{\nabla}\cdot\vec{v} = \left(\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z}, \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x}, \frac{\partial v_y}{\partial x} -\frac{\partial v_x}{\partial y}\right) $$

$$\Delta\psi = \frac{\partial^2 \psi_x}{\partial x^2}+ \frac{\partial^2 \psi_y}{\partial y^2}+ \frac{\partial^2 \psi_z}{\partial z^2} $$

Transformation to other coordinates
The general transformation relations to a new coordinate system $$(u_1, u_2, u_3) $$ are,

$$\mathrm{d}V = \frac{\mathrm{d}u_1}{U_1}\frac{\mathrm{d}u_2}{U_2}\frac{\mathrm{d}u_3}{U_3}$$

$$\vec{\nabla}\psi = \left(U_1\frac{\partial\psi}{\partial u_1}, U_2\frac{\partial \psi}{\partial u_2}, U_3\frac{\partial \psi}{\partial u_3}\right) $$

$$\vec{\nabla}\cdot\vec{v} = U_1U_2U_3\left(\frac{\partial}{\partial u_1}\frac{v_{u_1}}{U_2U_3}+ \frac{\partial}{\partial u_2}\frac{v_{u_2}}{U_1U_3}+ \frac{\partial}{\partial u_3}\frac{v_{u_3}}{U_1U_2}\right) $$$$\vec{\nabla}\times\vec{v} = \left(

U_2U_3(\frac{\partial}{\partial u_2}\frac{v_{u_3}}{U_3} - \frac{\partial}{\partial u_3}\frac{v_{u_2}}{U_2}),

U_1U_3(\frac{\partial}{\partial u_3}\frac{v_{u_1}}{U_1}-\frac{\partial}{\partial u_1}\frac{v_{u_3}}{U_3}),

U_1U_2(\frac{\partial}{\partial u_1}\frac{v_{u_2}}{U_2}-\frac{\partial}{\partial u_2}\frac{v_{u_1}}{U_1}) \right) $$$$\Delta\psi = U_1U_2U_3\left[ \frac{\partial}{\partial u_1}(\frac{U_1}{U_2U_3}\frac{\partial \psi}{\partial u_1}) + \frac{\partial}{\partial u_2}(\frac{U_2}{U_1U_3}\frac{\partial \psi}{\partial u_2}) + \frac{\partial}{\partial u_3}(\frac{U_3}{U_1U_2}\frac{\partial \psi}{\partial u_3}) \right] $$,

where

$$U_1^{-1} = \sqrt{\left(\frac{\partial x}{\partial u_1}\right)^2 + \left(\frac{\partial y}{\partial u_1}\right)^2 + \left(\frac{\partial z}{\partial u_1}\right)^2}, $$

$$U_2^{-1} = \sqrt{\left(\frac{\partial x}{\partial u_2}\right)^2 + \left(\frac{\partial y}{\partial u_2}\right)^2 + \left(\frac{\partial z}{\partial u_2}\right)^2}, $$

$$U_3^{-1} = \sqrt{\left(\frac{\partial x}{\partial u_3}\right)^2 + \left(\frac{\partial y}{\partial u_3}\right)^2 + \left(\frac{\partial z}{\partial u_3}\right)^2}, $$

$$v_{u_1} = v_x U_1 \frac{\partial x}{\partial u_1} + v_y U_1 \frac{\partial y}{\partial u_1} + v_z U_1 \frac{\partial z}{\partial u_1} $$

$$v_{u_2} = v_x U_2 \frac{\partial x}{\partial u_2} + v_y U_2 \frac{\partial y}{\partial u_2} + v_z U_2 \frac{\partial z}{\partial u_2} $$

$$v_{u_3} = v_x U_3 \frac{\partial x}{\partial u_3} + v_y U_3 \frac{\partial y}{\partial u_3} + v_z U_3 \frac{\partial z}{\partial u_3} $$

Cylindrical Coordinates
The cylindrical coordinates $$(u_1 = r, u_2 = \varphi, u_3=z) $$ are related to the cartesian coordinates by

$$(x, y, z) = (r\cos\varphi, r\sin\varphi, z) $$

$$\mathrm{d}V = r\ \mathrm{d}r\mathrm{d}\varphi\mathrm{d}z$$

$$\vec{\nabla}\psi = \left(\frac{\partial \psi}{\partial r}, \frac{1}{r}\frac{\partial \psi}{\partial \varphi}, \frac{\partial \psi}{\partial z}\right) $$

$$\vec{\nabla}\cdot\vec{v} = \frac{1}{r}\frac{\partial}{\partial r}(rv_r) + \frac{1}{r}\frac{\partial v_\varphi}{\partial\varphi}+ \frac{\partial v_z}{\partial z} $$

$$\vec{\nabla}\times\vec{v} = \left(

\frac{1}{r}\frac{\partial v_z}{\partial \varphi}-\frac{\partial v_\phi}{v_z},

\frac{\partial v_r}{\partial z}-\frac{\partial v_z}{\partial r},

\frac{1}{r}\frac{\partial}{\partial r}(rv_\phi)-\frac{1}{r}\frac{\partial v_r}{\partial \phi} \right) $$

$$\Delta\psi = \frac{\partial^2\psi}{\partial r^2} + \frac{1}{r}\frac{\partial \psi}{\partial r} + \frac{1}{r^2}\frac{\partial^2\psi}{\partial \varphi^2} + \frac{\partial^2\psi}{\partial z^2} $$

Spherical Polar Coordinates
The spherical polar coordinates $$(u_1 = r, u_2 = \theta, u_3=\varphi) $$, or simply the spherical coordinates, are particularly useful when the system in $$\mathrm{R}^3 $$ has a spherical symmetry, such as the motion of a particle under the influence of central forces.

$$(x, y, z) = (r\sin\theta\cos\varphi, r\sin\theta\sin\varphi, r\cos\theta) $$

$$U_1^{-1} = 1,\ U_2^{-1} = r,\ U_3^{-1} = r\sin\theta $$

$$\mathrm{d}V = r^2\sin\theta\ \mathrm{d}r\mathrm{d}\varphi\mathrm{d}\theta$$

$$\vec{\nabla}\psi = \left(\frac{\partial \psi}{\partial r}, \frac{1}{r}\frac{\partial \psi}{\partial \theta}, \frac{1}{r\sin\theta}\frac{\partial \psi}{\partial \varphi}\right) $$

$$\vec{\nabla}\cdot\vec{v} = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2v_r) + \frac{1}{r\sin\theta}\frac{\partial}{\partial\theta} (v_\theta \sin\theta)+ \frac{1}{r\sin\theta}\frac{\partial v_\varphi}{\partial \varphi} $$

$$\vec{\nabla}\times\vec{v} = \left(

\frac{1}{r\sin\theta}\left(\frac{\partial}{\partial \theta}(\sin\theta v_\varphi)-\frac{\partial v_\theta}{\partial \varphi}\right),

\frac{1}{r\sin\theta}\left(\frac{\partial v_r}{\partial \varphi}-\sin\theta\frac{\partial}{\partial r}(r v_\varphi)\right),

\frac{1}{r}\left(\frac{\partial}{\partial r}(r v_\theta)-\frac{\partial v_r}{\partial \theta}\right) \right) $$

$$\Delta\psi = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial \psi}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 \psi}{\partial \varphi^2} $$