Cylindrical coordinate system

Point in cylindrical coordinates.

A cylindrical coordinate system is a three-dimensional coordinate system , which is an extension of the polar coordinate system by adding a third coordinate (usually denoted $z$ ${\ displaystyle z}$ $z$ ), which sets the height of the point above the plane.

Point $P$ ${\ displaystyle P}$ $P$ given as $(\rho ,\;\varphi ,\;z)$ ${\ displaystyle (\ rho, \; \ varphi, \; z)}$ $(\ rho, \; \ varphi, \; z)$ . In terms of a rectangular coordinate system :

$\rho \geqslant 0$ ${\ displaystyle \ rho \ geqslant 0}$ $\ rho \ geqslant 0$ - distance from $O$ ${\ displaystyle O}$ $O$ before $P^{'}$ ${\ displaystyle P '}$ $P '$ , orthogonal projection of a point $P$ ${\ displaystyle P}$ $P$ to the plane $X Y$ ${\ displaystyle XY}$ $Xy$ . Or the same as the distance from $P$ ${\ displaystyle P}$ $P$ to the axis $Z$ ${\ displaystyle Z}$ $Z$ .
$0\leqslant \varphi <360^{\circ }$ ${\ displaystyle 0 \ leqslant \ varphi <360 ^ {\ circ}}$ $0 \ leqslant \ varphi <360 ^ {\ circ}$ - angle between axis $X$ ${\ displaystyle X}$ $X$ and segment $O P^{'}$ ${\ displaystyle OP '}$ $OP '$ .
$z$ ${\ displaystyle z}$ $z$ equal to applicate point $P$ ${\ displaystyle P}$ $P$ .

When used in the physical sciences and technology, the international standard ISO 31-11 recommends the use of the notation $(\rho ,\;\varphi ,\;z)$ ${\ displaystyle (\ rho, \; \ varphi, \; z)}$ $(\ rho, \; \ varphi, \; z)$ .

Cylindrical coordinates are useful when analyzing surfaces that are symmetrical about an axis if the axis $Z$ ${\ displaystyle Z}$ $Z$ take as the axis of symmetry. For example, an infinitely long round cylinder (cylindrical surface) in rectangular coordinates has the equation $x^{2}+y^{2}=c^{2}$ ${\ displaystyle x ^ {2} + y ^ {2} = c ^ {2}}$ $x ^ {2} + y ^ {2} = c ^ {2}$ , and in cylindrical - a very simple equation $\rho =c$ ${\ displaystyle \ rho = c}$ $\ rho = c$ . From here comes the name "cylindrical" for a given coordinate system.

Content

Transition to other coordinate systems

2 points in cylindrical coordinates.

Since a cylindrical coordinate system is only one of many three-dimensional coordinate systems, there are laws for the transformation of coordinates between a cylindrical coordinate system and other systems.

Cartesian coordinate system

The unit vectors of the cylindrical coordinate system are related to the Cartesian unit vectors by the following relations:

${\begin{cases}{\vec {e}}_{\rho }=\cos \varphi {\vec {e}}_{x}+\sin \varphi {\vec {e}}_{y},\\{\vec {e}}_{\varphi }=-\sin \varphi {\vec {e}}_{x}+\cos \varphi {\vec {e}}_{y},\\{\vec {e}}_{z}={\vec {e}}_{z},\end{cases}}$ ${\ displaystyle {\ begin {cases} {\ vec {e}} _ {\ rho} = \ cos \ varphi {\ vec {e}} _ {x} + \ sin \ varphi {\ vec {e}} _ {y}, \\ {\ vec {e}} _ {\ varphi} = - \ sin \ varphi {\ vec {e}} _ {x} + \ cos \ varphi {\ vec {e}} _ {y }, \\ {\ vec {e}} _ {z} = {\ vec {e}} _ {z}, \ end {cases}}}$

and form the right three:

${\begin{cases}{\vec {e}}_{\rho }\times {\vec {e}}_{\varphi }={\vec {e}}_{z},\\{\vec {e}}_{z}\times {\vec {e}}_{\rho }={\vec {e}}_{\varphi },\\{\vec {e}}_{\varphi }\times {\vec {e}}_{z}={\vec {e}}_{\rho }.\end{cases}}$ ${\ displaystyle {\ begin {cases} {\ vec {e}} _ {\ rho} \ times {\ vec {e}} _ {\ varphi} = {\ vec {e}} _ {z}, \\ {\ vec {e}} _ {z} \ times {\ vec {e}} _ {\ rho} = {\ vec {e}} _ {\ varphi}, \\ {\ vec {e}} _ { \ varphi} \ times {\ vec {e}} _ {z} = {\ vec {e}} _ {\ rho}. \ end {cases}}}$

The inverse relations have the form:

${\begin{cases}{\vec {e}}_{x}=\cos \varphi {\vec {e}}_{\rho }-\sin \varphi {\vec {e}}_{\varphi },\\{\vec {e}}_{y}=\sin \varphi {\vec {e}}_{\rho }+\cos \varphi {\vec {e}}_{\varphi },\\{\vec {e}}_{z}={\vec {e}}_{z}.\end{cases}}$ ${\ displaystyle {\ begin {cases} {\ vec {e}} _ {x} = \ cos \ varphi {\ vec {e}} _ {\ rho} - \ sin \ varphi {\ vec {e}} _ {\ varphi}, \\ {\ vec {e}} _ {y} = \ sin \ varphi {\ vec {e}} _ {\ rho} + \ cos \ varphi {\ vec {e}} _ {\ varphi}, \\ {\ vec {e}} _ {z} = {\ vec {e}} _ {z}. \ end {cases}}}$

The law of transformation of coordinates from cylindrical to Cartesian:

{\begin{cases}x=\rho \cos \varphi ,\\y=\rho \sin \varphi ,\\z=z.\end{cases}}

{\ displaystyle {\ begin {cases} x = \ rho \ cos \ varphi, \\ y = \ rho \ sin \ varphi, \\ z = z. \ end {cases}}}

The law of transformation of coordinates from Cartesian to cylindrical:

{\begin{cases}\rho ={\sqrt {x^{2}+y^{2}}},\\\varphi =\mathrm {arctg} \left({\dfrac {y}{x}}\right),\\z=z.\end{cases}}

{\ displaystyle {\ begin {cases} \ rho = {\ sqrt {x ^ {2} + y ^ {2}}}, \\\ varphi = \ mathrm {arctg} \ left ({\ dfrac {y} { x}} \ right), \\ z = z. \ end {cases}}}

Jacobian is equal to:

J=\rho .

{\ displaystyle J = \ rho.}

Differential characteristics

The cylindrical coordinates are orthogonal, therefore, the metric tensor has the diagonal form in them:

g_{ij}={\begin{pmatrix}1&0&0\\0&\rho ^{2}&0\\0&0&1\end{pmatrix}},\quad g^{ij}={\begin{pmatrix}1&0&0\\0&1/\rho ^{2}&0\\0&0&1\end{pmatrix}}.

{\ displaystyle g_ {ij} = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & \ rho ^ {2} & 0 \\ 0 & 0 & 1 \ end {pmatrix}}, \ quad g ^ {ij} = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 / \ rho ^ {2} & 0 \\ 0 & 0 & 1 \ end {pmatrix}}.}

Differential squared curve length

ds^{2}=d\rho ^{2}+\rho ^{2}\,d\varphi ^{2}+dz^{2}.

{\ displaystyle ds ^ {2} = d \ rho ^ {2} + \ rho ^ {2} \, d \ varphi ^ {2} + dz ^ {2}.}

Lame coefficients have the form:

H_{\rho }=1,\quad H_{\varphi }=\rho ,\quad H_{z}=1.

{\ displaystyle H _ {\ rho} = 1, \ quad H _ {\ varphi} = \ rho, \ quad H_ {z} = 1.}

Christoffel Symbols $\{\rho ,\;\varphi ,\;z\}$ ${\ displaystyle \ {\ rho, \; \ varphi, \; z \}}$ :

\Gamma _{22}^{1}=-\rho ,\quad \Gamma _{21}^{2}=\Gamma _{12}^{2}={\frac {1}{\rho }}.

{\ displaystyle \ Gamma _ {22} ^ {1} = - \ rho, \ quad \ Gamma _ {21} ^ {2} = \ Gamma _ {12} ^ {2} = {\ frac {1} {\ rho}}.}

The rest are zero.

Differential Operators

Gradient in a cylindrical coordinate system:

\mathrm {grad} \,\psi ={\vec {e}}_{\rho }{\frac {\partial \psi }{\partial \rho }}+{\vec {e}}_{\varphi }{\frac {1}{\rho }}{\frac {\partial \psi }{\partial \varphi }}+{\vec {e}}_{z}{\frac {\partial \psi }{\partial z}}.

{\ displaystyle \ mathrm {grad} \, \ psi = {\ vec {e}} _ {\ rho} {\ frac {\ partial \ psi} {\ partial \ rho}} + {\ vec {e}} _ {\ varphi} {\ frac {1} {\ rho}} {\ frac {\ partial \ psi} {\ partial \ varphi}} + {\ vec {e}} _ {z} {\ frac {\ partial \ psi} {\ partial z}}.}

Divergence in a cylindrical coordinate system:

\mathrm {div} \,{\vec {a}}={\frac {1}{\rho }}{\frac {\partial \rho a_{\rho }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial a_{\varphi }}{\partial \varphi }}+{\frac {\partial a_{z}}{\partial z}}.

{\ displaystyle \ mathrm {div} \, {\ vec {a}} = {\ frac {1} {\ rho}} {\ frac {\ partial \ rho a _ {\ rho}} {\ partial \ rho}} + {\ frac {1} {\ rho}} {\ frac {\ partial a _ {\ varphi}} {\ partial \ varphi}} + {\ frac {\ partial a_ {z}} {\ partial z}}. }

Rotor in a cylindrical coordinate system:

\mathrm {rot} \,{\vec {a}}=\mathrm {det} {\begin{pmatrix}{\frac {1}{\rho }}{\vec {e}}_{\rho }&{\vec {e}}_{\varphi }&{\frac {1}{\rho }}{\vec {e}}_{z}\\{\frac {\partial }{\partial \rho }}&{\frac {\partial }{\partial \varphi }}&{\frac {\partial }{\partial z}}\\a_{\rho }&\rho a_{\varphi }&\ a_{z}\end{pmatrix}}={\vec {e}}_{\rho }\left({\frac {1}{\rho }}{\frac {\partial a_{z}}{\partial \varphi }}-{\frac {\partial a_{\varphi }}{\partial z}}\right)+{\vec {e}}_{\varphi }\left({\frac {\partial a_{\rho }}{\partial z}}-{\frac {\partial a_{z}}{\partial \rho }}\right)+{\vec {e}}_{z}\left({\frac {1}{\rho }}{\frac {\partial \rho a_{\varphi }}{\partial \rho }}-{\frac {1}{\rho }}{\frac {\partial a_{\rho }}{\partial \varphi }}\right).

{\ displaystyle \ mathrm {rot} \, {\ vec {a}} = \ mathrm {det} {\ begin {pmatrix} {\ frac {1} {\ rho}} {\ vec {e}} _ {\ rho} & {\ vec {e}} _ {\ varphi} & {\ frac {1} {\ rho}} {\ vec {e}} _ {z} \\ {\ frac {\ partial} {\ partial \ rho}} & {\ frac {\ partial} {\ partial \ varphi}} & {\ frac {\ partial} {\ partial z}} \\ a _ {\ rho} & \ rho a _ {\ varphi} & \ a_ {z} \ end {pmatrix}} = {\ vec {e}} _ {\ rho} \ left ({\ frac {1} {\ rho}} {\ frac {\ partial a_ {z}} {\ partial \ varphi}} - {\ frac {\ partial a _ {\ varphi}} {\ partial z}} \ right) + {\ vec {e}} _ {\ varphi} \ left ({\ frac {\ partial a_ {\ rho}} {\ partial z}} - {\ frac {\ partial a_ {z}} {\ partial \ rho}} \ right) + {\ vec {e}} _ {z} \ left ({\ frac {1} {\ rho}} {\ frac {\ partial \ rho a _ {\ varphi}} {\ partial \ rho}} - {\ frac {1} {\ rho}} {\ frac {\ partial a_ { \ rho}} {\ partial \ varphi}} \ right).}

Expressions for the radius vector , velocity and acceleration in cylindrical coordinates

$r(t)=\rho {\vec {e}}_{\rho }+z{\vec {e}}_{z}$ ${\ displaystyle r (t) = \ rho {\ vec {e}} _ {\ rho} + z {\ vec {e}} _ {z}}$

${\dot {r}}(t)={\dot {\rho }}{\vec {e}}_{\rho }+\rho {\dot {\varphi }}{\vec {e}}_{\varphi }+{\dot {z}}{\vec {e}}_{z}$ ${\ displaystyle {\ dot {r}} (t) = {\ dot {\ rho}} {\ vec {e}} _ {\ rho} + \ rho {\ dot {\ varphi}} {\ vec {e }} _ {\ varphi} + {\ dot {z}} {\ vec {e}} _ {z}}$

${\ddot {r}}(t)=({\ddot {\rho }}-\rho {\dot {\varphi }}^{2}){\vec {e}}_{\rho }+(2{\dot {\rho }}{\dot {\varphi }}+{\ddot {\varphi }}\rho ){\vec {e}}_{\varphi }+{\ddot {z}}{\vec {e}}_{z}$ ${\ displaystyle {\ ddot {r}} (t) = ({\ ddot {\ rho}} - \ rho {\ dot {\ varphi}} ^ {2}) {\ vec {e}} _ {\ rho } + (2 {\ dot {\ rho}} {\ dot {\ varphi}} + {\ ddot {\ varphi}} \ rho) {\ vec {e}} _ {\ varphi} + {\ ddot {z }} {\ vec {e}} _ {z}}$

Literature

Khalilov V.R., Chizhov G.A., Dynamics of classical systems: Textbook. allowance. - M.: Publishing House of Moscow State University, 1993 .-- 352 p.