Toroidal coordinate system

A toroidal coordinate system is an orthogonal coordinate system in space whose coordinate surfaces are tori, spheres, and half-planes. This coordinate system can be obtained by rotating a two-dimensional bipolar coordinate system around an axis equidistant from the foci of the bipolar system.

Content

1 Definition
2 Properties
- 2.1 Coordinate surfaces
- 2.2 Differential characteristics
3 Type of differential operators in toroidal coordinates
4 Differential equations in toroidal coordinates
5 Literature
6 References

Definition

Toroidal coordinate system $(\alpha ,\beta ,\varphi )$ ${\ displaystyle (\ alpha, \ beta, \ varphi)}$ $(\alpha ,\beta ,\varphi )$ determined by the transition formulas from these coordinates to Cartesian coordinates :

x={\frac {c\,\mathrm {sh} \,\alpha \cos \varphi }{\mathrm {ch} \,\alpha -\cos \beta }}\quad \quad y={\frac {c\,\mathrm {sh} \,\alpha \sin \varphi }{\mathrm {ch} \,\alpha -\cos \beta }}\quad \quad z={\frac {c\sin \beta }{\mathrm {ch} \,\alpha -\cos \beta }}

{\ displaystyle x = {\ frac {c \, \ mathrm {sh} \, \ alpha \ cos \ varphi} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} \ quad \ quad y = { \ frac {c \, \ mathrm {sh} \, \ alpha \ sin \ varphi} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} \ quad \ quad z = {\ frac {c \ sin \ beta} {\ mathrm {ch} \, \ alpha - \ cos \ beta}}}

x={\frac {c\,\mathrm {sh} \,\alpha \cos \varphi }{\mathrm {ch} \,\alpha -\cos \beta }}\quad \quad y={\frac {c\,\mathrm {sh} \,\alpha \sin \varphi }{\mathrm {ch} \,\alpha -\cos \beta }}\quad \quad z={\frac {c\sin \beta }{\mathrm {ch} \,\alpha -\cos \beta }}

,

Where $c>0$ ${\ displaystyle c> 0}$ $c>0$ - a scale factor that must be fixed to select a specific toroidal coordinate system, $0\leqslant \alpha <\infty ,-\pi <\beta \leqslant \pi ,-\pi <\varphi \leqslant \pi$ ${\ displaystyle 0 \ leqslant \ alpha <\ infty, - \ pi <\ beta \ leqslant \ pi, - \ pi <\ varphi \ leqslant \ pi}$ $0\leqslant \alpha <\infty ,-\pi <\beta \leqslant \pi ,-\pi <\varphi \leqslant \pi$ .

Properties

Coordinate surfaces

$\alpha =\mathrm {const}$ ${\ displaystyle \ alpha = \ mathrm {const}}$ $\alpha =\mathrm {const}$ - tori

({\sqrt {x^{2}+y^{2}}}-c\,\mathrm {cth} \,\alpha )^{2}+z^{2}=\left({\frac {c}{\mathrm {sh} \,\alpha }}\right)^{2}

{\ displaystyle ({\ sqrt {x ^ {2} + y ^ {2}}} - c \, \ mathrm {cth} \, \ alpha) ^ {2} + z ^ {2} = \ left ({ \ frac {c} {\ mathrm {sh} \, \ alpha}} \ right) ^ {2}}

({\sqrt {x^{2}+y^{2}}}-c\,\mathrm {cth} \,\alpha )^{2}+z^{2}=\left({\frac {c}{\mathrm {sh} \,\alpha }}\right)^{2}

,

$\beta =\mathrm {const}$ ${\ displaystyle \ beta = \ mathrm {const}}$ $\beta =\mathrm {const}$ - spheres

(z-c\,\mathrm {ctg} \,\beta )^{2}+x^{2}+y^{2}=\left({\frac {c}{\sin \beta }}\right)^{2}

{\ displaystyle (zc \, \ mathrm {ctg} \, \ beta) ^ {2} + x ^ {2} + y ^ {2} = \ left ({\ frac {c} {\ sin \ beta}} \ right) ^ {2}}

(z-c\,\mathrm {ctg} \,\beta )^{2}+x^{2}+y^{2}=\left({\frac {c}{\sin \beta }}\right)^{2}

,

$\varphi =\mathrm {const}$ ${\ displaystyle \ varphi = \ mathrm {const}}$ $\varphi =\mathrm {const}$ - half planes

{\frac {x}{\cos \varphi }}={\frac {y}{\sin \varphi }}

{\ displaystyle {\ frac {x} {\ cos \ varphi}} = {\ frac {y} {\ sin \ varphi}}}

{\frac {x}{\cos \varphi }}={\frac {y}{\sin \varphi }}

.

Differential characteristics

The metric tensor in toroidal coordinates has the form:

g_{ij}={\begin{pmatrix}{\frac {c^{2}}{(\mathrm {ch} \,\alpha -\cos \beta )^{2}}}&0&0\\0&{\frac {c^{2}}{(\mathrm {ch} \,\alpha -\cos \beta )^{2}}}&0\\0&0&{\frac {c^{2}\mathrm {sh} ^{2}\alpha }{(\mathrm {ch} \,\alpha -\cos \beta )^{2}}}\end{pmatrix}},\quad g^{ij}={\begin{pmatrix}{\frac {(\mathrm {ch} \,\alpha -\cos \beta )^{2}}{c^{2}}}&0&0\\0&{\frac {(\mathrm {ch} \,\alpha -\cos \beta )^{2}}{c^{2}}}&0\\0&0&{\frac {(\mathrm {ch} \,\alpha -\cos \beta )^{2}}{c^{2}\,\mathrm {sh} ^{2}\,\alpha }}\end{pmatrix}}.

{\ displaystyle g_ {ij} = {\ begin {pmatrix} {\ frac {c ^ {2}} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {2}}} & 0 & 0 \\ 0 & {\ frac {c ^ {2}} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {2}}} & 0 \\ 0 & 0 & {\ frac {c ^ {2} \ mathrm { sh} ^ {2} \ alpha} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {2}}} \ end {pmatrix}}, \ quad g ^ {ij} = {\ begin {pmatrix} {\ frac {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {2}} {c ^ {2}}} & 0 & 0 \\ 0 & {\ frac {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {2}} {c ^ {2}}} & 0 \\ 0 & 0 & {\ frac {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {2 }} {c ^ {2} \, \ mathrm {sh} ^ {2} \, \ alpha}} \ end {pmatrix}}.}

g_{ij}={\begin{pmatrix}{\frac {c^{2}}{(\mathrm {ch} \,\alpha -\cos \beta )^{2}}}&0&0\\0&{\frac {c^{2}}{(\mathrm {ch} \,\alpha -\cos \beta )^{2}}}&0\\0&0&{\frac {c^{2}\mathrm {sh} ^{2}\alpha }{(\mathrm {ch} \,\alpha -\cos \beta )^{2}}}\end{pmatrix}},\quad g^{ij}={\begin{pmatrix}{\frac {(\mathrm {ch} \,\alpha -\cos \beta )^{2}}{c^{2}}}&0&0\\0&{\frac {(\mathrm {ch} \,\alpha -\cos \beta )^{2}}{c^{2}}}&0\\0&0&{\frac {(\mathrm {ch} \,\alpha -\cos \beta )^{2}}{c^{2}\,\mathrm {sh} ^{2}\,\alpha }}\end{pmatrix}}.

It is diagonal, since the toroidal coordinate system is orthogonal .

Linear element square:

ds^{2}={\frac {c^{2}}{(\mathrm {ch} \,\alpha -\cos \beta )^{2}}}(d\alpha ^{2}+d\beta ^{2}+\mathrm {sh} ^{2}\alpha \,d\varphi ^{2})

{\ displaystyle ds ^ {2} = {\ frac {c ^ {2}} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {2}}} (d \ alpha ^ {2} + d \ beta ^ {2} + \ mathrm {sh} ^ {2} \ alpha \, d \ varphi ^ {2})}

.

Square Element Square:

dS^{2}={\frac {c^{4}}{(\mathrm {ch} \,\alpha -\cos \beta )^{4}}}((d\alpha \,d\beta )^{2}+\mathrm {sh} ^{2}\alpha (d\alpha \,d\varphi )^{2}+\mathrm {sh} ^{2}\alpha (d\beta \,d\varphi )^{2})

{\ displaystyle dS ^ {2} = {\ frac {c ^ {4}} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {4}}} ((d \ alpha \, d \ beta) ^ {2} + \ mathrm {sh} ^ {2} \ alpha (d \ alpha \, d \ varphi) ^ {2} + \ mathrm {sh} ^ {2} \ alpha (d \ beta \ , d \ varphi) ^ {2})}

.

Volume Element:

dV={\frac {c^{3}\mathrm {sh} \,\alpha }{(\mathrm {ch} \,\alpha -\cos \beta )^{3}}}d\alpha \,d\beta \,d\varphi

{\ displaystyle dV = {\ frac {c ^ {3} \ mathrm {sh} \, \ alpha} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {3}}} d \ alpha \, d \ beta \, d \ varphi}

.

Lame Odds :

h_{\alpha }=h_{\beta }={\frac {c}{\mathrm {ch} \,\alpha -\cos \beta }},\quad h_{\varphi }={\frac {c\,\mathrm {sh} \,\alpha }{\mathrm {ch} \,\alpha -\cos \beta }}

{\ displaystyle h _ {\ alpha} = h _ {\ beta} = {\ frac {c} {\ mathrm {ch} \, \ alpha - \ cos \ beta}}, \ quad h _ {\ varphi} = {\ frac {c \, \ mathrm {sh} \, \ alpha} {\ mathrm {ch} \, \ alpha - \ cos \ beta}}}

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Jacobian :

{\frac {\partial (x,y,z)}{\partial (\alpha ,\beta ,\varphi )}}={\frac {c^{3}\mathrm {sh} \,\alpha }{(\mathrm {ch} \,\alpha -\cos \beta )^{3}}}

{\ displaystyle {\ frac {\ partial (x, y, z)} {\ partial (\ alpha, \ beta, \ varphi)}} = {\ frac {c ^ {3} \ mathrm {sh} \, \ alpha} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {3}}}}

.

Kristoffel's symbols of the second kind:

\Gamma _{ij}^{1}={\begin{pmatrix}0&-{\frac {\sin \beta }{\mathrm {ch} \,\alpha -\cos \beta }}&0\\-{\frac {\sin \beta }{\mathrm {ch} \,\alpha -\cos \beta }}&{\frac {\mathrm {sh} \,\alpha }{\mathrm {ch} \,\alpha -\cos \beta }}&0\\0&0&{\frac {\mathrm {sh} \,\alpha (\mathrm {ch} \,\alpha \cos \beta -1)}{\mathrm {ch} \,\alpha -\cos \beta }}\end{pmatrix}},

{\ displaystyle \ Gamma _ {ij} ^ {1} = {\ begin {pmatrix} 0 & - {\ frac {\ sin \ beta} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} & 0 \ \ - {\ frac {\ sin \ beta} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} & {\ frac {\ mathrm {sh} \, \ alpha} {\ mathrm {ch} \ , \ alpha - \ cos \ beta}} & 0 \\ 0 & 0 & {\ frac {\ mathrm {sh} \, \ alpha (\ mathrm {ch} \, \ alpha \ cos \ beta -1)} {\ mathrm {ch } \, \ alpha - \ cos \ beta}} \ end {pmatrix}},}

\Gamma _{ij}^{2}={\begin{pmatrix}{\frac {\sin \beta }{\mathrm {ch} \,\alpha -\cos \beta }}&-{\frac {\mathrm {sh} \,\alpha }{\mathrm {ch} \,\alpha -\cos \beta }}&0\\-{\frac {\mathrm {sh} \,\alpha }{\mathrm {ch} \,\alpha -\cos \beta }}&0&0\\0&0&{\frac {\mathrm {sh} ^{2}\alpha \sin \beta }{\mathrm {ch} \,\alpha -\cos \beta }}\end{pmatrix}},

{\ displaystyle \ Gamma _ {ij} ^ {2} = {\ begin {pmatrix} {\ frac {\ sin \ beta} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} & - {\ frac {\ mathrm {sh} \, \ alpha} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} & 0 \\ - {\ frac {\ mathrm {sh} \, \ alpha} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} & 0 & 0 \\ 0 & 0 & {\ frac {\ mathrm {sh} ^ {2} \ alpha \ sin \ beta} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} \ end {pmatrix}},}

\Gamma _{ij}^{3}={\begin{pmatrix}0&0&-{\frac {1}{(\mathrm {ch} \,\alpha -\cos \beta )\mathrm {sh} ^{2}\alpha }}\\0&0&-{\frac {\sin \beta }{\mathrm {ch} \,\alpha -\cos \beta }}\\-{\frac {1}{(\mathrm {ch} \,\alpha -\cos \beta )\mathrm {sh} ^{2}\alpha }}&-{\frac {\sin \beta }{\mathrm {ch} \,\alpha -\cos \beta }}&0\end{pmatrix}}.

{\ displaystyle \ Gamma _ {ij} ^ {3} = {\ begin {pmatrix} 0 & 0 & - {\ frac {1} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) \ mathrm {sh} ^ {2} \ alpha}} \\ 0 & 0 & - {\ frac {\ sin \ beta} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} \\ - {\ frac {1} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) \ mathrm {sh} ^ {2} \ alpha}} & - {\ frac {\ sin \ beta} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} & 0 \ end {pmatrix}}.}

Type of differential operators in toroidal coordinates

The gradient of the scalar function in toroidal coordinates is given by the following expression:

\operatorname {grad} \,U(\alpha ,\;\beta ,\;\varphi )={\frac {\mathrm {ch} \,\alpha -\cos \beta }{c}}\left({\frac {\partial U}{\partial \alpha }}{\vec {e}}_{\alpha }+{\frac {\partial U}{\partial \beta }}{\vec {e}}_{\beta }+{\frac {1}{\mathrm {sh} \,\alpha }}{\frac {\partial U}{\partial \varphi }}{\vec {e}}_{\varphi }\right).

{\ displaystyle \ operatorname {grad} \, U (\ alpha, \; \ beta, \; \ varphi) = {\ frac {\ mathrm {ch} \, \ alpha - \ cos \ beta} {c}} \ left ({\ frac {\ partial U} {\ partial \ alpha}} {\ vec {e}} _ {\ alpha} + {\ frac {\ partial U} {\ partial \ beta}} {\ vec {e }} _ {\ beta} + {\ frac {1} {\ mathrm {sh} \, \ alpha}} {\ frac {\ partial U} {\ partial \ varphi}} {\ vec {e}} _ { \ varphi} \ right).}

Divergence of the vector field:

\ \operatorname {div} \mathbf {F} ={\frac {(\mathrm {ch} \,\alpha -\cos \beta )^{2}}{c^{2}\,\mathrm {sh} \,\alpha }}\left({\frac {\partial F_{\alpha }}{\partial \alpha }}+{\frac {\partial F_{\beta }}{\partial \beta }}+\,\mathrm {sh} \,\alpha {\frac {\partial F_{\varphi }}{\partial \varphi }}\right)-{\frac {\mathrm {ch} \,\alpha -\cos \beta }{c^{2}\,\mathrm {sh} \,\alpha }}(F_{\alpha }\,\mathrm {sh} \,\alpha -F_{\beta }\sin \beta )

{\ displaystyle \ \ operatorname {div} \ mathbf {F} = {\ frac {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {2}} {c ^ {2} \, \ mathrm {sh} \, \ alpha}} \ left ({\ frac {\ partial F _ {\ alpha}} {\ partial \ alpha}} + {\ frac {\ partial F _ {\ beta}} {\ partial \ beta} } + \, \ mathrm {sh} \, \ alpha {\ frac {\ partial F _ {\ varphi}} {\ partial \ varphi}} \ right) - {\ frac {\ mathrm {ch} \, \ alpha - \ cos \ beta} {c ^ {2} \, \ mathrm {sh} \, \ alpha}} (F _ {\ alpha} \, \ mathrm {sh} \, \ alpha -F _ {\ beta} \ sin \ beta)}

Laplace operator :

\Delta u={\frac {(\mathrm {ch} \,\alpha -\cos \beta )^{3}}{c^{2}\,\mathrm {sh} \,\alpha }}\left({\frac {\partial }{\partial \alpha }}\left({\frac {\,\mathrm {sh} \,\alpha }{\mathrm {ch} \,\alpha -\cos \beta }}{\frac {\partial u}{\partial \alpha }}\right)+{\frac {\partial }{\partial \beta }}\left({\frac {\,\mathrm {sh} \,\alpha }{\mathrm {ch} \,\alpha -\cos \beta }}{\frac {\partial u}{\partial \beta }}\right)+{\frac {1}{(\mathrm {ch} \,\alpha -\cos \beta )\,\mathrm {sh} \,\alpha }}{\frac {\partial ^{2}u}{\partial \varphi ^{2}}}\right)

{\ displaystyle \ Delta u = {\ frac {(\ mathrm {ch} \, \ alpha - \ cos \ beta) ^ {3}} {c ^ {2} \, \ mathrm {sh} \, \ alpha} } \ left ({\ frac {\ partial} {\ partial \ alpha}} \ left ({\ frac {\, \ mathrm {sh} \, \ alpha} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} {\ frac {\ partial u} {\ partial \ alpha}} \ right) + {\ frac {\ partial} {\ partial \ beta}} \ left ({\ frac {\, \ mathrm {sh } \, \ alpha} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} {\ frac {\ partial u} {\ partial \ beta}} \ right) + {\ frac {1} {( \ mathrm {ch} \, \ alpha - \ cos \ beta) \, \ mathrm {sh} \, \ alpha}} {\ frac {\ partial ^ {2} u} {\ partial \ varphi ^ {2}} } \ right)}

Differential equations in toroidal coordinates

The Laplace equation in toroidal coordinates has the form:

\left({\frac {\partial }{\partial \alpha }}\left({\frac {\,\mathrm {sh} \,\alpha }{\mathrm {ch} \,\alpha -\cos \beta }}{\frac {\partial u}{\partial \alpha }}\right)+{\frac {\partial }{\partial \beta }}\left({\frac {\,\mathrm {sh} \,\alpha }{\mathrm {ch} \,\alpha -\cos \beta }}{\frac {\partial u}{\partial \beta }}\right)+{\frac {1}{(\mathrm {ch} \,\alpha -\cos \beta )\,\mathrm {sh} \,\alpha }}{\frac {\partial ^{2}u}{\partial \varphi ^{2}}}\right)=0

{\ displaystyle \ left ({\ frac {\ partial} {\ partial \ alpha}} \ left ({\ frac {\, \ mathrm {sh} \, \ alpha} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} {\ frac {\ partial u} {\ partial \ alpha}} \ right) + {\ frac {\ partial} {\ partial \ beta}} \ left ({\ frac {\, \ mathrm {sh} \, \ alpha} {\ mathrm {ch} \, \ alpha - \ cos \ beta}} {\ frac {\ partial u} {\ partial \ beta}} \ right) + {\ frac {1} {(\ mathrm {ch} \, \ alpha - \ cos \ beta) \, \ mathrm {sh} \, \ alpha}} {\ frac {\ partial ^ {2} u} {\ partial \ varphi ^ {2 }}} \ right) = 0}

It is convenient to search for a solution in the form:

u=v{\sqrt {2\mathrm {ch} \alpha -2\cos \beta }}

{\ displaystyle u = v {\ sqrt {2 \ mathrm {ch} \ alpha -2 \ cos \ beta}}}

,

then the equation for the function $v$ ${\ displaystyle v}$ :

v_{\alpha \alpha }+v_{\beta \beta }+v_{\alpha }\mathrm {cth} \,\alpha +{\frac {1}{4}}v+{\frac {1}{\mathrm {sh} ^{2}\alpha }}v_{\varphi \varphi }=0

{\ displaystyle v _ {\ alpha \ alpha} + v _ {\ beta \ beta} + v _ {\ alpha} \ mathrm {cth} \, \ alpha + {\ frac {1} {4}} v + {\ frac {1 } {\ mathrm {sh} ^ {2} \ alpha}} v _ {\ varphi \ varphi} = 0}

.

Then you can separate the variables:

v=A(\alpha )B(\beta )\Phi (\varphi )

{\ displaystyle v = A (\ alpha) B (\ beta) \ Phi (\ varphi)}

.

The result is a system:

{\begin{cases}A''+\,\mathrm {cth} \,\alpha \,A'+\left({\frac {1}{4}}-{\frac {k_{\varphi }^{2}}{\mathrm {sh} ^{2}\alpha }}-k_{\beta }^{2}\right)A=0\\B''+k_{\beta }^{2}B=0\\\Phi ''+k_{\varphi }^{2}\Phi =0\end{cases}}

{\ displaystyle {\ begin {cases} A '' + \, \ mathrm {cth} \, \ alpha \, A '+ \ left ({\ frac {1} {4}} - {\ frac {k _ {\ varphi} ^ {2}} {\ mathrm {sh} ^ {2} \ alpha}} - k _ {\ beta} ^ {2} \ right) A = 0 \\ B '' + k _ {\ beta} ^ { 2} B = 0 \\\ Phi '' + k _ {\ varphi} ^ {2} \ Phi = 0 \ end {cases}}}

In the case of the Helmholtz equation in toroidal coordinates, the variables are not divided.

Literature

Korn G., Korn T. Chapter 6. Systems of curvilinear coordinates. 6.5 Formulas for orthogonal coordinate systems // Handbook of mathematics (for scientists and engineers). - M .: Nauka, 1973. - S. 195. - 832 p.
Morse F. M., Feshbach G. Chapter 5. Ordinary differential equations. The table of separating coordinates for three dimensions // Methods of theoretical physics. - M .: Publishing house of foreign literature, 1958. - T. 1. - S. 622. - 930 p.
Tikhonov A.N., Samarsky A.A. Part IV. Formulas, tables, graphs. IV. Various orthogonal coordinate systems // Equations of mathematical physics. - 7th ed. - M .: Publishing House of Moscow State University; Nauka, 2004 .-- S. 732-733. - 798 p. - ISBN 5-211-04843-1 .

Links

Weisstein, Eric W. Toroidal Coordinates on Wolfram MathWorld .