Elliptical coordinate system

Elliptical coordinates - a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolas . For two tricks $F_{1}$ ${\ displaystyle F_ {1}}$ $F_ {1}$ and $F_{2}$ ${\ displaystyle F_ {2}}$ $F_ {2}$ points are usually taken $-c$ ${\ displaystyle -c}$ ${\ displaystyle -c}$ and $+c$ ${\ displaystyle + c}$ ${\ displaystyle + c}$ on axis $X$ ${\ displaystyle X}$ $X$ Cartesian coordinate system.

Content

Basic Definition

Elliptical coordinates $(\mu ,\;\nu )$ ${\ displaystyle (\ mu, \; \ nu)}$ usually determined by the rule:

\left\{{\begin{matrix}x=c\,\mathrm {ch} \,\mu \cos \nu \\y=c\,\mathrm {sh} \,\mu \sin \nu \end{matrix}}\right.

{\ displaystyle \ left \ {{\ begin {matrix} x = c \, \ mathrm {ch} \, \ mu \ cos \ nu \\ y = c \, \ mathrm {sh} \, \ mu \ sin \ nu \ end {matrix}} \ right.}

Where $\mu \geqslant 0$ ${\ displaystyle \ mu \ geqslant 0}$ , $\nu \in [0,\;2\pi )$ ${\ displaystyle \ nu \ in [0, \; 2 \ pi)}$ .

In this way, a family of confocal ellipses and hyperbolas is defined. Trigonometric identity

{\frac {x^{2}}{c^{2}\,\mathrm {ch} ^{2}\,\mu }}+{\frac {y^{2}}{c^{2}\,\mathrm {sh} ^{2}\,\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1

{\ displaystyle {\ frac {x ^ {2}} {c ^ {2} \, \ mathrm {ch} ^ {2} \, \ mu}} + {\ frac {y ^ {2}} {c ^ {2} \, \ mathrm {sh} ^ {2} \, \ mu}} = \ cos ^ {2} \ nu + \ sin ^ {2} \ nu = 1}

shows level lines $\mu$ ${\ displaystyle \ mu}$ are ellipses , and an identity from hyperbolic geometry

{\frac {x^{2}}{c^{2}\cos ^{2}\nu }}-{\frac {y^{2}}{c^{2}\sin ^{2}\nu }}=\mathrm {ch} ^{2}\,\mu -\mathrm {sh} ^{2}\,\mu =1

{\ displaystyle {\ frac {x ^ {2}} {c ^ {2} \ cos ^ {2} \ nu}} - {\ frac {y ^ {2}} {c ^ {2} \ sin ^ { 2} \ nu}} = \ mathrm {ch} ^ {2} \, \ mu - \ mathrm {sh} ^ {2} \, \ mu = 1}

shows level lines $\nu$ ${\ displaystyle \ nu}$ are hyperbolas .

Lame Odds

Lame coefficients for elliptical coordinates $(\mu ,\;\nu )$ ${\ displaystyle (\ mu, \; \ nu)}$ are equal

H_{\mu }=H_{\nu }=c{\sqrt {(\mathrm {ch} \,\mu \,\sin \,\nu )^{2}+(\mathrm {sh} \,\mu \,\cos \,\nu )^{2}}}=c{\sqrt {\mathrm {sh} ^{2}\,\mu +\sin ^{2}\nu }}.

{\ displaystyle H _ {\ mu} = H _ {\ nu} = c {\ sqrt {(\ mathrm {ch} \, \ mu \, \ sin \, \ nu) ^ {2} + (\ mathrm {sh} \, \ mu \, \ cos \, \ nu) ^ {2}}} = c {\ sqrt {\ mathrm {sh} ^ {2} \, \ mu + \ sin ^ {2} \ nu}}. }

The identities for the double angle allow us to bring them to the form

H_{\mu }=H_{\nu }=c{\sqrt {{\frac {1}{2}}(\mathrm {ch} \,2\mu -\cos 2\nu }}).

{\ displaystyle H _ {\ mu} = H _ {\ nu} = c {\ sqrt {{\ frac {1} {2}} (\ mathrm {ch} \, 2 \ mu - \ cos 2 \ nu}}) .}

The area element is:

dS=c^{2}(\mathrm {sh} ^{2}\,\mu +\sin ^{2}\nu )\,d\mu \,d\nu ,

{\ displaystyle dS = c ^ {2} (\ mathrm {sh} ^ {2} \, \ mu + \ sin ^ {2} \ nu) \, d \ mu \, d \ nu,}

and Laplacian is

\nabla ^{2}\Phi ={\frac {1}{c^{2}(\mathrm {sh} ^{2}\,\mu +\sin ^{2}\nu )}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right).

{\ displaystyle \ nabla ^ {2} \ Phi = {\ frac {1} {c ^ {2} (\ mathrm {sh} ^ {2} \, \ mu + \ sin ^ {2} \ nu)}} \ left ({\ frac {\ partial ^ {2} \ Phi} {\ partial \ mu ^ {2}}} + {\ frac {\ partial ^ {2} \ Phi} {\ partial \ nu ^ {2} }} \ right).}

Other differential operators can be obtained by substituting the Lame coefficients in general formulas for orthogonal coordinates. For example, a scalar field gradient $\Phi (\mu ,\;\nu )$ ${\ displaystyle \ Phi (\ mu, \; \ nu)}$ is recorded:

\mathrm {grad} \,\Phi ={\frac {1}{H_{\mu }}}{\frac {\partial \Phi }{\partial \mu }}\mathbf {e} _{\mu }+{\frac {1}{H_{\nu }}}{\frac {\partial \Phi }{\partial \nu }}\mathbf {e} _{\nu },

{\ displaystyle \ mathrm {grad} \, \ Phi = {\ frac {1} {H _ {\ mu}}} {\ frac {\ partial \ Phi} {\ partial \ mu}} \ mathbf {e} _ { \ mu} + {\ frac {1} {H _ {\ nu}}} {\ frac {\ partial \ Phi} {\ partial \ nu}} \ mathbf {e} _ {\ nu},}

Where

\mathbf {e} _{\mu }=c(\mathrm {sh} \,\mu \cos \nu ,\,\mathrm {ch} \,\mu \sin \nu )

{\ displaystyle \ mathbf {e} _ {\ mu} = c (\ mathrm {sh} \, \ mu \ cos \ nu, \, \ mathrm {ch} \, \ mu \ sin \ nu)}

,

\mathbf {e} _{\nu }=c(-\mathrm {ch} \,\mu \sin \nu ,\,\mathrm {sh} \,\mu \cos \nu )

{\ displaystyle \ mathbf {e} _ {\ nu} = c (- \ mathrm {ch} \, \ mu \ sin \ nu, \, \ mathrm {sh} \, \ mu \ cos \ nu)}

.

Another definition

Sometimes another more geometrically intuitive definition of elliptic coordinates is used. $(\sigma ,\;\tau )$ ${\ displaystyle (\ sigma, \; \ tau)}$ :

\left\{{\begin{matrix}\sigma =\mathrm {ch} \,\mu \\\tau =\cos \nu \end{matrix}}\right.

{\ displaystyle \ left \ {{\ begin {matrix} \ sigma = \ mathrm {ch} \, \ mu \\\ tau = \ cos \ nu \ end {matrix}} \ right.}

Thus, the level lines $\sigma$ ${\ displaystyle \ sigma}$ are ellipses and level lines $\tau$ ${\ displaystyle \ tau}$ are hyperbolas. Wherein

\tau \in [-1,\;1],\quad \sigma \geqslant 1.

{\ displaystyle \ tau \ in [-1, \; 1], \ quad \ sigma \ geqslant 1.}

Coordinates $(\sigma ,\;\tau )$ ${\ displaystyle (\ sigma, \; \ tau)}$ have a simple connection with the distances to the tricks $F_{1}$ ${\ displaystyle F_ {1}}$ and $F_{2}$ ${\ displaystyle F_ {2}}$ . For any point on the plane

\left\{{\begin{matrix}d_{1}+d_{2}=2c\sigma \\d_{1}-d_{2}=2c\tau \end{matrix}}\right.

{\ displaystyle \ left \ {{\ begin {matrix} d_ {1} + d_ {2} = 2c \ sigma \\ d_ {1} -d_ {2} = 2c \ tau \ end {matrix}} \ right. }

Where $d_{1},\;d_{2}$ ${\ displaystyle d_ {1}, \; d_ {2}}$ - distance to tricks $F_{1},\;F_{2}$ ${\ displaystyle F_ {1}, \; F_ {2}}$ respectively.

In this way:

\left\{{\begin{matrix}d_{1}=c(\sigma +\tau )\\d_{2}=c(\sigma -\tau )\end{matrix}}\right.

{\ displaystyle \ left \ {{\ begin {matrix} d_ {1} = c (\ sigma + \ tau) \\ d_ {2} = c (\ sigma - \ tau) \ end {matrix}} \ right. }

Recall that $F_{1}$ ${\ displaystyle F_ {1}}$ and $F_{2}$ ${\ displaystyle F_ {2}}$ are at points $x=-c$ ${\ displaystyle x = -c}$ and $x=+c$ ${\ displaystyle x = + c}$ respectively.

The disadvantage of this coordinate system is that it does not display one-to-one on Cartesian coordinates:

\left\{{\begin{matrix}x=c\sigma \tau \\y^{2}=c^{2}(\sigma ^{2}-1)(1-\tau ^{2})\end{matrix}}\right.

{\ displaystyle \ left \ {{\ begin {matrix} x = c \ sigma \ tau \\ y ^ {2} = c ^ {2} (\ sigma ^ {2} -1) (1- \ tau ^ { 2}) \ end {matrix}} \ right.}

Lame Odds

Lame coefficients for alternative elliptical coordinates $(\sigma ,\;\tau )$ ${\ displaystyle (\ sigma, \; \ tau)}$ equal to:

h_{\sigma }=c{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}};

{\ displaystyle h _ {\ sigma} = c {\ sqrt {\ frac {\ sigma ^ {2} - \ tau ^ {2}} {\ sigma ^ {2} -1}}}}

h_{\tau }=c{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{1-\tau ^{2}}}}.

{\ displaystyle h _ {\ tau} = c {\ sqrt {\ frac {\ sigma ^ {2} - \ tau ^ {2}} {1- \ tau ^ {2}}}}.

The area element is

dA=c^{2}{\frac {\sigma ^{2}-\tau ^{2}}{\sqrt {(\sigma ^{2}-1)(1-\tau ^{2})}}}\,d\sigma \,d\tau ,

{\ displaystyle dA = c ^ {2} {\ frac {\ sigma ^ {2} - \ tau ^ {2}} {\ sqrt {(\ sigma ^ {2} -1) (1- \ tau ^ {2 })}}} \, d \ sigma \, d \ tau,}

and Laplacian is

\nabla ^{2}\Phi ={\frac {1}{c^{2}(\sigma ^{2}-\tau ^{2})}}\left[{\sqrt {\sigma ^{2}-1}}{\frac {\partial }{\partial \sigma }}\left({\sqrt {\sigma ^{2}-1}}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\sqrt {1-\tau ^{2}}}{\frac {\partial }{\partial \tau }}\left({\sqrt {1-\tau ^{2}}}{\frac {\partial \Phi }{\partial \tau }}\right)\right].

{\ displaystyle \ nabla ^ {2} \ Phi = {\ frac {1} {c ^ {2} (\ sigma ^ {2} - \ tau ^ {2})}} left [{\ sqrt {\ sigma ^ {2} -1}} {\ frac {\ partial} {\ partial \ sigma}} \ left ({\ sqrt {\ sigma ^ {2} -1}} {\ frac {\ partial \ Phi} {\ partial \ sigma}} \ right) + {\ sqrt {1- \ tau ^ {2}}} {\ frac {\ partial} {\ partial \ tau}} \ left ({\ sqrt {1- \ tau ^ { 2}}} {\ frac {\ partial \ Phi} {\ partial \ tau}} \ right) \ right].}

Other differential operators can be obtained by substituting the Lame coefficients in general formulas for orthogonal coordinates.

Literature

Korn G., Korn T. Handbook of mathematics (for scientists and engineers). - M .: Nauka, 1974.- 832 p.

Elliptical coordinate system

Content

Basic Definition

Lame Odds

Another definition

Lame Odds

Literature

See also

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