# Dynamics/Kinematics/Coordinate Systems/Spherical

## Introduction

A spherical coordinate system is also useful for describing motion.

Material taken from Vector fields in cylindrical and spherical coordinates

## Position

Vectors are defined in spherical coordinates by (r, θ, φ), where

• r is the length of the vector,
• θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and
• φ is the angle between the projection of the vector onto the X-Y-plane and the positive X-axis (0 ≤ φ < 2π).

(r, θ, φ) is given in Cartesian coordinates by:

${\displaystyle {\begin{bmatrix}r\\\theta \\\phi \end{bmatrix}}={\begin{bmatrix}{\sqrt {x^{2}+y^{2}+z^{2}}}\\\arccos(z/r)\\\arctan(y/x)\end{bmatrix}},\ \ \ 0\leq \theta \leq \pi ,\ \ \ 0\leq \phi <2\pi ,}$

or inversely by:

${\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}r\sin \theta \cos \phi \\r\sin \theta \sin \phi \\r\cos \theta \end{bmatrix}}.}$

Any vector field can be written in terms of the unit vectors as:

${\displaystyle \mathbf {A} =A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}} =A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}}$

The spherical unit vectors are related to the cartesian unit vectors by:

${\displaystyle {\begin{bmatrix}{\boldsymbol {\hat {r}}}\\{\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}\end{bmatrix}}={\begin{bmatrix}\sin \theta \cos \phi &\sin \theta \sin \phi &\cos \theta \\\cos \theta \cos \phi &\cos \theta \sin \phi &-\sin \theta \\-\sin \phi &\cos \phi &0\end{bmatrix}}{\begin{bmatrix}\mathbf {\hat {x}} \\\mathbf {\hat {y}} \\\mathbf {\hat {z}} \end{bmatrix}}}$

Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

So the cartesian unit vectors are related to the spherical unit vectors by:

${\displaystyle {\begin{bmatrix}\mathbf {\hat {x}} \\\mathbf {\hat {y}} \\\mathbf {\hat {z}} \end{bmatrix}}={\begin{bmatrix}\sin \theta \cos \phi &\cos \theta \cos \phi &-\sin \phi \\\sin \theta \sin \phi &\cos \theta \sin \phi &\cos \phi \\\cos \theta &-\sin \theta &0\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {\hat {r}}}\\{\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}\end{bmatrix}}}$