Accelerator Physics/Coordinate Systems

 Cartesian coordinates

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

$\overrightarrow{\mathrm{\nabla }}\psi =(\frac{\partial \psi }{\partial x},\frac{\partial \psi }{\partial y},\frac{\partial \psi }{\partial z})$

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

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

$\mathrm{\Delta }\psi =\frac{{\partial }^2{\psi }_x}{\partial x^2}+\frac{{\partial }^2{\psi }_y}{\partial y^2}+\frac{{\partial }^2{\psi }_z}{\partial z^2}$

Transformation to other coordinates

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

$\mathrm{d}V=\frac{\mathrm{d}{u}_1}{U_1}\frac{\mathrm{d}{u}_2}{U_2}\frac{\mathrm{d}{u}_3}{U_3}$

$\overrightarrow{\mathrm{\nabla }}\psi =(U_1\frac{\partial \psi }{\partial u_1},U_2\frac{\partial \psi }{\partial u_2},U_3\frac{\partial \psi }{\partial u_3})$

$\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{v}=U_1{U}_2{U}_3(\frac{\partial }{\partial u_1}\frac{v_{u_1}}{U_2{U}_3}+\frac{\partial }{\partial u_2}\frac{v_{u_2}}{U_1{U}_3}+\frac{\partial }{\partial u_3}\frac{v_{u_3}}{U_1{U}_2})$$\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{v}=(U_2{U}_3(\frac{\partial }{\partial u_2}\frac{v_{u_3}}{U_3}-\frac{\partial }{\partial u_3}\frac{v_{u_2}}{U_2}),U_1{U}_3(\frac{\partial }{\partial u_3}\frac{v_{u_1}}{U_1}-\frac{\partial }{\partial u_1}\frac{v_{u_3}}{U_3}),U_1{U}_2(\frac{\partial }{\partial u_1}\frac{v_{u_2}}{U_2}-\frac{\partial }{\partial u_2}\frac{v_{u_1}}{U_1}))$$\mathrm{\Delta }\psi =U_1{U}_2{U}_3[\frac{\partial }{\partial u_1}(\frac{U_1}{U_2{U}_3}\frac{\partial \psi }{\partial u_1})+\frac{\partial }{\partial u_2}(\frac{U_2}{U_1{U}_3}\frac{\partial \psi }{\partial u_2})+\frac{\partial }{\partial u_3}(\frac{U_3}{U_1{U}_2}\frac{\partial \psi }{\partial u_3})]$,

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 $(u_1=r,u_2=\phi ,u_3=z)$ are related to the cartesian coordinates by

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

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

$\overrightarrow{\mathrm{\nabla }}\psi =(\frac{\partial \psi }{\partial r},\frac{1}{r}\frac{\partial \psi }{\partial \phi },\frac{\partial \psi }{\partial z})$

$\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{v}=\frac{1}{r}\frac{\partial }{\partial r}(r{v}_r)+\frac{1}{r}\frac{\partial v_{\phi }}{\partial \phi }+\frac{\partial v_z}{\partial z}$

$\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{v}=(\frac{1}{r}\frac{\partial v_z}{\partial \phi }-\frac{\partial v_{\varphi }}{v_z},\frac{\partial v_r}{\partial z}-\frac{\partial v_z}{\partial r},\frac{1}{r}\frac{\partial }{\partial r}(r{v}_{\varphi })-\frac{1}{r}\frac{\partial v_r}{\partial \varphi })$

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

Spherical Polar Coordinates

Spherical coordinates in most of the physics conventions. Notice that this is different than the mathematics convention in which $\theta$ and $\phi$ are swapped compared to the figure shown.

The spherical polar coordinates $(u_1=r,u_2=\theta ,u_3=\phi )$, 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.

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

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

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

$\overrightarrow{\mathrm{\nabla }}\psi =(\frac{\partial \psi }{\partial r},\frac{1}{r}\frac{\partial \psi }{\partial \theta },\frac{1}{r\sin \theta }\frac{\partial \psi }{\partial \phi })$

$\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{v}=\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{v}_r)+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(v_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial v_{\phi }}{\partial \phi }$

$\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{v}=(\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(\sin \theta v_{\phi })-\frac{\partial v_{\theta }}{\partial \phi }),\frac{1}{r\sin \theta }(\frac{\partial v_r}{\partial \phi }-\sin \theta \frac{\partial }{\partial r}(r{v}_{\phi })),\frac{1}{r}(\frac{\partial }{\partial r}(r{v}_{\theta })-\frac{\partial v_r}{\partial \theta }))$

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