Kegelkoordinaten

Kegelkoordinaten sind orthogonale Koordinaten, in denen ein Punkt des dreidimensionalen Raums durch Angabe der Lage auf einer Kugel und zwei elliptischen Kegeln bestimmt wird, siehe Bild.

Kegelkoordinaten (englisch conical coordinates^{[1] :37[2] :659}) erlauben immer eine Trennung der Veränderlichen in der Laplace- und Helmholtz-Gleichung,^{[1] :8} was deren Lösung vereinfacht. Kegelkoordinaten bieten sich zur Lösung von Randwertaufgaben dort an, wo die Ränder des Gebiets kugel- oder kegelförmig sind. Zur Anpassung an diese Ränder dienen zwei Parameter, im Bild b und c, die die Form der #Koordinatenflächen beeinflussen. Kegelkoordinaten wurden auf mehrere verschiedene Arten definiert.^[3]

Sie sind nicht zu verwechseln mit den nicht-orthogonalen Polarkoordinaten gleichen Namens.

Die kartesischen Koordinaten ${\displaystyle (x,y,z)}${\displaystyle (x,y,z)} berechnen sich aus den Kegelkoordinaten ${\displaystyle r,\theta ,\lambda \in \mathbb {R} ,r,\theta >0}${\displaystyle r,\theta ,\lambda \in \mathbb {R} ,r,\theta >0} bei gegebenem ${\displaystyle b,c\in \mathbb {R} }${\displaystyle b,c\in \mathbb {R} } gemäß:^{[1] :37}

${\displaystyle {\vec {r}}:={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\frac {r}{bc{\sqrt {c^{2}-b^{2}}}}}{\begin{pmatrix}\theta \lambda {\sqrt {c^{2}-b^{2}}}\\c{\sqrt {(\theta ^{2}-b^{2})(b^{2}-\lambda ^{2})}}\\b{\sqrt {(c^{2}-\theta ^{2})(c^{2}-\lambda ^{2})}}\end{pmatrix}}}${\displaystyle {\vec {r}}:={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\frac {r}{bc{\sqrt {c^{2}-b^{2}}}}}{\begin{pmatrix}\theta \lambda {\sqrt {c^{2}-b^{2}}}\\c{\sqrt {(\theta ^{2}-b^{2})(b^{2}-\lambda ^{2})}}\\b{\sqrt {(c^{2}-\theta ^{2})(c^{2}-\lambda ^{2})}}\end{pmatrix}}}

Damit das möglich ist, muss

${\displaystyle 0\leq \lambda ^{2}<b^{2}<\theta ^{2}<c^{2}}${\displaystyle 0\leq \lambda ^{2}<b^{2}<\theta ^{2}<c^{2}}

sein. Die Koordinatenflächen bestehen aus einer Kugel (r=const., rot im Bild oben),

${\displaystyle x^{2}+y^{2}+z^{2}=r^{2}}${\displaystyle x^{2}+y^{2}+z^{2}=r^{2}}

einem elliptischen Kegel um die z-Achse (θ=const., blau im Bild oben),

${\displaystyle {\frac {x^{2}}{\theta ^{2}}}+{\frac {y^{2}}{\theta ^{2}-b^{2}}}-{\frac {z^{2}}{c^{2}-\theta ^{2}}}=0}${\displaystyle {\frac {x^{2}}{\theta ^{2}}}+{\frac {y^{2}}{\theta ^{2}-b^{2}}}-{\frac {z^{2}}{c^{2}-\theta ^{2}}}=0}

und einem elliptischen Kegel um die x-Achse (λ=const., gelb im Bild oben),

${\displaystyle {\frac {x^{2}}{\lambda ^{2}}}-{\frac {y^{2}}{b^{2}-\lambda ^{2}}}-{\frac {z^{2}}{c^{2}-\lambda ^{2}}}=0}${\displaystyle {\frac {x^{2}}{\lambda ^{2}}}-{\frac {y^{2}}{b^{2}-\lambda ^{2}}}-{\frac {z^{2}}{c^{2}-\lambda ^{2}}}=0}

Aus obigen drei Gleichungen können die Koordinatenquadrate bestimmt werden:

${\displaystyle {\begin{aligned}r^{2}=&x^{2}+y^{2}+z^{2}\\\theta ^{2}=&{\frac {b^{2}(r^{2}-y^{2})+c^{2}(r^{2}-z^{2})+{\sqrt {[b^{2}(r^{2}-y^{2})+c^{2}(r^{2}-z^{2})]^{2}-(2bcrx)^{2}}}}{2r^{2}}}\\\lambda ^{2}=&{\frac {b^{2}(r^{2}-y^{2})+c^{2}(r^{2}-z^{2})-{\sqrt {[b^{2}(r^{2}-y^{2})+c^{2}(r^{2}-z^{2})]^{2}-(2bcrx)^{2}}}}{2r^{2}}}\end{aligned}}}${\displaystyle {\begin{aligned}r^{2}=&x^{2}+y^{2}+z^{2}\\\theta ^{2}=&{\frac {b^{2}(r^{2}-y^{2})+c^{2}(r^{2}-z^{2})+{\sqrt {[b^{2}(r^{2}-y^{2})+c^{2}(r^{2}-z^{2})]^{2}-(2bcrx)^{2}}}}{2r^{2}}}\\\lambda ^{2}=&{\frac {b^{2}(r^{2}-y^{2})+c^{2}(r^{2}-z^{2})-{\sqrt {[b^{2}(r^{2}-y^{2})+c^{2}(r^{2}-z^{2})]^{2}-(2bcrx)^{2}}}}{2r^{2}}}\end{aligned}}}

Eine alternative Formulierung^{[2] :659} benutzt die Koordinaten

${\displaystyle {\begin{aligned}\xi _{1}=&r,\;\xi _{2}={\frac {\sqrt {b^{2}-\lambda ^{2}}}{c}}=\alpha ,円{\rm {cn}}(\mu ,\alpha ),\;\xi _{3}={\frac {\sqrt {\theta ^{2}-b^{2}}}{c}}=\beta ,円{\rm {cn}}(\nu ,\beta )\\x=&{\frac {\xi _{1}}{\alpha }}{\sqrt {(\alpha ^{2}-\xi _{2}^{2})(\alpha ^{2}+\xi _{3}^{2})}}=\xi _{1},円{\rm {sn}}(\mu ,\alpha ),円{\rm {dn}}(\nu ,\beta )\\y=&{\frac {\xi _{1}}{\beta }}{\sqrt {(\beta ^{2}+\xi _{2}^{2})(\beta ^{2}-\xi _{3}^{2})}}=\xi _{1},円{\rm {dn}}(\mu ,\alpha ),円{\rm {sn}}(\nu ,\beta )\\z=&{\frac {\xi _{1}\xi _{2}\xi _{3}}{\alpha \beta }}=\xi _{1},円{\rm {cn}}(\mu ,\alpha ),円{\rm {cn}}(\nu ,\beta )\end{aligned}}}${\displaystyle {\begin{aligned}\xi _{1}=&r,\;\xi _{2}={\frac {\sqrt {b^{2}-\lambda ^{2}}}{c}}=\alpha ,円{\rm {cn}}(\mu ,\alpha ),\;\xi _{3}={\frac {\sqrt {\theta ^{2}-b^{2}}}{c}}=\beta ,円{\rm {cn}}(\nu ,\beta )\\x=&{\frac {\xi _{1}}{\alpha }}{\sqrt {(\alpha ^{2}-\xi _{2}^{2})(\alpha ^{2}+\xi _{3}^{2})}}=\xi _{1},円{\rm {sn}}(\mu ,\alpha ),円{\rm {dn}}(\nu ,\beta )\\y=&{\frac {\xi _{1}}{\beta }}{\sqrt {(\beta ^{2}+\xi _{2}^{2})(\beta ^{2}-\xi _{3}^{2})}}=\xi _{1},円{\rm {dn}}(\mu ,\alpha ),円{\rm {sn}}(\nu ,\beta )\\z=&{\frac {\xi _{1}\xi _{2}\xi _{3}}{\alpha \beta }}=\xi _{1},円{\rm {cn}}(\mu ,\alpha ),円{\rm {cn}}(\nu ,\beta )\end{aligned}}}

mit den drei grundlegenden Jacobischen Funktionen sinus– sn, cosinus– cn bzw. delta amplitudinis dn, dem elliptischen Modul ${\displaystyle \alpha ={\sqrt {b/c}}}${\displaystyle \alpha ={\sqrt {b/c}}}, dem komplementären Parameter ${\displaystyle \beta ={\sqrt {1-\alpha ^{2}}}}${\displaystyle \beta ={\sqrt {1-\alpha ^{2}}}} und beliebigen Variablen μ,ν∈R.

Die Koordinatenflächen sind hier eine Kugel und elliptische Kegel um die x- und y-Achse:

${\displaystyle {\begin{aligned}{\frac {x^{2}+y^{2}+z^{2}}{\xi _{1}^{2}}}=&1\\{\frac {x^{2}}{\alpha ^{2}-\xi _{2}^{2}}}-{\frac {y^{2}}{\beta ^{2}+\xi _{2}^{2}}}-{\frac {z^{2}}{\xi _{2}^{2}}}=&0\\{\frac {x^{2}}{\alpha ^{2}+\xi _{3}^{2}}}-{\frac {y^{2}}{\beta ^{2}-\xi _{3}^{2}}}+{\frac {z^{2}}{\xi _{3}^{2}}}=&0\end{aligned}}}${\displaystyle {\begin{aligned}{\frac {x^{2}+y^{2}+z^{2}}{\xi _{1}^{2}}}=&1\\{\frac {x^{2}}{\alpha ^{2}-\xi _{2}^{2}}}-{\frac {y^{2}}{\beta ^{2}+\xi _{2}^{2}}}-{\frac {z^{2}}{\xi _{2}^{2}}}=&0\\{\frac {x^{2}}{\alpha ^{2}+\xi _{3}^{2}}}-{\frac {y^{2}}{\beta ^{2}-\xi _{3}^{2}}}+{\frac {z^{2}}{\xi _{3}^{2}}}=&0\end{aligned}}}

Hieraus lässt sich für die in diesem Artikel benutzte Formulierung

${\displaystyle {\begin{aligned}\theta =&c,円{\rm {dn}}(\nu ,\beta ),\;\lambda =b,円{\rm {sn}}(\mu ,\alpha )\\x=&r,円{\rm {sn}}(\mu ,\alpha ),円{\rm {dn}}(\nu ,\beta ),\;y=r,円{\rm {cn}}(\mu ,\alpha ),円{\rm {cn}}(\nu ,\beta ),\;z=r,円{\rm {dn}}(\mu ,\alpha ),円{\rm {sn}}(\nu ,\beta )\end{aligned}}}${\displaystyle {\begin{aligned}\theta =&c,円{\rm {dn}}(\nu ,\beta ),\;\lambda =b,円{\rm {sn}}(\mu ,\alpha )\\x=&r,円{\rm {sn}}(\mu ,\alpha ),円{\rm {dn}}(\nu ,\beta ),\;y=r,円{\rm {cn}}(\mu ,\alpha ),円{\rm {cn}}(\nu ,\beta ),\;z=r,円{\rm {dn}}(\mu ,\alpha ),円{\rm {sn}}(\nu ,\beta )\end{aligned}}}

ableiten.

Die kovarianten Basisvektoren sind mit ${\displaystyle {\vec {r}}=(x,y,z)^{\top }}${\displaystyle {\vec {r}}=(x,y,z)^{\top }}:

${\displaystyle {\vec {g}}_{r}:={\frac {\partial {\vec {r}}}{\partial r}}={\frac {1}{r}}{\begin{pmatrix}x\\y\\z\end{pmatrix}},\;{\vec {g}}_{\theta }:={\frac {\partial {\vec {r}}}{\partial \theta }}={\begin{pmatrix}{\frac {x}{\theta }}\\{\frac {y\theta }{\theta ^{2}-b^{2}}}\\{\frac {-z\theta }{c^{2}-\theta ^{2}}}\end{pmatrix}},\;{\vec {g}}_{\lambda }:={\frac {\partial {\vec {r}}}{\partial \lambda }}={\begin{pmatrix}{\frac {x}{\lambda }}\\{\frac {y\lambda }{\lambda ^{2}-b^{2}}}\\{\frac {-z\lambda }{c^{2}-\lambda ^{2}}}\end{pmatrix}}}${\displaystyle {\vec {g}}_{r}:={\frac {\partial {\vec {r}}}{\partial r}}={\frac {1}{r}}{\begin{pmatrix}x\\y\\z\end{pmatrix}},\;{\vec {g}}_{\theta }:={\frac {\partial {\vec {r}}}{\partial \theta }}={\begin{pmatrix}{\frac {x}{\theta }}\\{\frac {y\theta }{\theta ^{2}-b^{2}}}\\{\frac {-z\theta }{c^{2}-\theta ^{2}}}\end{pmatrix}},\;{\vec {g}}_{\lambda }:={\frac {\partial {\vec {r}}}{\partial \lambda }}={\begin{pmatrix}{\frac {x}{\lambda }}\\{\frac {y\lambda }{\lambda ^{2}-b^{2}}}\\{\frac {-z\lambda }{c^{2}-\lambda ^{2}}}\end{pmatrix}}}

die, wie es sein muss, senkrecht zueinander sind, und deren Beträge die metrischen Faktoren sind:

${\displaystyle {\begin{aligned}h_{r}:=|{\vec {g}}_{r}|=1,\quad h_{\theta }:=|{\vec {g}}_{\theta }|=r{\sqrt {\frac {\theta ^{2}-\lambda ^{2}}{(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})}}},\\h_{\lambda }:=|{\vec {g}}_{\lambda }|=r{\sqrt {\frac {\theta ^{2}-\lambda ^{2}}{(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2})}}}\end{aligned}}}${\displaystyle {\begin{aligned}h_{r}:=|{\vec {g}}_{r}|=1,\quad h_{\theta }:=|{\vec {g}}_{\theta }|=r{\sqrt {\frac {\theta ^{2}-\lambda ^{2}}{(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})}}},\\h_{\lambda }:=|{\vec {g}}_{\lambda }|=r{\sqrt {\frac {\theta ^{2}-\lambda ^{2}}{(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2})}}}\end{aligned}}}

Das Orthonormalsystem ist dann

${\displaystyle {\begin{aligned}{\hat {c}}_{1}=&{\frac {1}{bc{\sqrt {c^{2}-b^{2}}}}}{\begin{pmatrix}\theta \lambda {\sqrt {c^{2}-b^{2}}}\\c{\sqrt {(\theta ^{2}-b^{2})(b^{2}-\lambda ^{2})}}\\b{\sqrt {(c^{2}-\theta ^{2})(c^{2}-\lambda ^{2})}}\end{pmatrix}}/()\\{\hat {c}}_{2}=&{\frac {1}{bc{\sqrt {(c^{2}-b^{2})(\theta ^{2}-\lambda ^{2})}}}}{\begin{pmatrix}\lambda {\sqrt {c^{2}-b^{2}}}{\sqrt {(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})}}\\\theta c{\sqrt {(c^{2}-\theta ^{2})(b^{2}-\lambda ^{2})}}\\-\theta b{\sqrt {(\theta ^{2}-b^{2})(c^{2}-\lambda ^{2})}}\end{pmatrix}}\\{\hat {c}}_{3}=&{\frac {1}{bc{\sqrt {(c^{2}-b^{2})(\theta ^{2}-\lambda ^{2})}}}}{\begin{pmatrix}\theta {\sqrt {c^{2}-b^{2}}}{\sqrt {(c^{2}-\lambda ^{2})(b^{2}-\lambda ^{2})}}\\-\lambda c{\sqrt {(\theta ^{2}-b^{2})(c^{2}-\lambda ^{2})}}\\-\lambda b{\sqrt {(c^{2}-\theta ^{2})(b^{2}-\lambda ^{2})}}\end{pmatrix}}\end{aligned}}}${\displaystyle {\begin{aligned}{\hat {c}}_{1}=&{\frac {1}{bc{\sqrt {c^{2}-b^{2}}}}}{\begin{pmatrix}\theta \lambda {\sqrt {c^{2}-b^{2}}}\\c{\sqrt {(\theta ^{2}-b^{2})(b^{2}-\lambda ^{2})}}\\b{\sqrt {(c^{2}-\theta ^{2})(c^{2}-\lambda ^{2})}}\end{pmatrix}}/()\\{\hat {c}}_{2}=&{\frac {1}{bc{\sqrt {(c^{2}-b^{2})(\theta ^{2}-\lambda ^{2})}}}}{\begin{pmatrix}\lambda {\sqrt {c^{2}-b^{2}}}{\sqrt {(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})}}\\\theta c{\sqrt {(c^{2}-\theta ^{2})(b^{2}-\lambda ^{2})}}\\-\theta b{\sqrt {(\theta ^{2}-b^{2})(c^{2}-\lambda ^{2})}}\end{pmatrix}}\\{\hat {c}}_{3}=&{\frac {1}{bc{\sqrt {(c^{2}-b^{2})(\theta ^{2}-\lambda ^{2})}}}}{\begin{pmatrix}\theta {\sqrt {c^{2}-b^{2}}}{\sqrt {(c^{2}-\lambda ^{2})(b^{2}-\lambda ^{2})}}\\-\lambda c{\sqrt {(\theta ^{2}-b^{2})(c^{2}-\lambda ^{2})}}\\-\lambda b{\sqrt {(c^{2}-\theta ^{2})(b^{2}-\lambda ^{2})}}\end{pmatrix}}\end{aligned}}}

Das Linien- und Volumenelement lauten^{[1] :38}

${\displaystyle {\begin{aligned}{\rm {d}}{\vec {r}}=&{\vec {g}}_{r},円{\rm {d}}r+{\vec {g}}_{\theta },円{\rm {d}}\theta +{\vec {g}}_{\lambda },円{\rm {d}}\lambda \\{\rm {d}}s^{2}:=&|{\rm {d}}{\vec {r}}|^{2}={\rm {d}}r^{2}+{\frac {r^{2}(\theta ^{2}-\lambda ^{2})}{(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})}}{\rm {d}}\theta ^{2}+{\frac {r^{2}(\theta ^{2}-\lambda ^{2})}{(c^{2}-\lambda ^{2})(b^{2}-\lambda ^{2})}}{\rm {d}}\lambda ^{2}\\{\rm {d}}V=&{\frac {-r^{4}(\theta ^{2}-\lambda ^{2})}{(c^{2}-b^{2})bcyz}}{\rm {d}}r,円{\rm {d}}\theta ,円{\rm {d}}\lambda \end{aligned}}}${\displaystyle {\begin{aligned}{\rm {d}}{\vec {r}}=&{\vec {g}}_{r},円{\rm {d}}r+{\vec {g}}_{\theta },円{\rm {d}}\theta +{\vec {g}}_{\lambda },円{\rm {d}}\lambda \\{\rm {d}}s^{2}:=&|{\rm {d}}{\vec {r}}|^{2}={\rm {d}}r^{2}+{\frac {r^{2}(\theta ^{2}-\lambda ^{2})}{(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})}}{\rm {d}}\theta ^{2}+{\frac {r^{2}(\theta ^{2}-\lambda ^{2})}{(c^{2}-\lambda ^{2})(b^{2}-\lambda ^{2})}}{\rm {d}}\lambda ^{2}\\{\rm {d}}V=&{\frac {-r^{4}(\theta ^{2}-\lambda ^{2})}{(c^{2}-b^{2})bcyz}}{\rm {d}}r,円{\rm {d}}\theta ,円{\rm {d}}\lambda \end{aligned}}}

Die Basisvektoren bilden demnach ein Rechtssystem, wo das Produkt bcyz im Nenner des Volumenelements negativ ist.

Wegen der länglichen Ausdrücke für die metrischen Faktoren, wird auf die allgemeine Darstellung der Operatoren Gradient, Divergenz und Rotation eines Vektorfeldes im Hauptartikel verwiesen.

Der Laplace-Operator ist:^{[1] :38}

${\displaystyle {\begin{aligned}\Delta f=&{\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+\dots \\&\dots +{\frac {1}{r^{2}(\theta ^{2}-\lambda ^{2})}}{\Bigg \{}(\theta ^{2}-b^{2})(c^{2}-\theta ^{2}){\frac {\partial ^{2}f}{\partial \theta ^{2}}}-\theta (2\theta ^{2}-b^{2}-c^{2}){\frac {\partial f}{\partial \theta }}+\dots \\&\qquad \qquad \qquad \qquad \;\dots +(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2}){\frac {\partial ^{2}f}{\partial \lambda ^{2}}}+\lambda (2\lambda ^{2}-b^{2}-c^{2}){\frac {\partial f}{\partial \lambda }}{\Bigg \}}\end{aligned}}}${\displaystyle {\begin{aligned}\Delta f=&{\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+\dots \\&\dots +{\frac {1}{r^{2}(\theta ^{2}-\lambda ^{2})}}{\Bigg \{}(\theta ^{2}-b^{2})(c^{2}-\theta ^{2}){\frac {\partial ^{2}f}{\partial \theta ^{2}}}-\theta (2\theta ^{2}-b^{2}-c^{2}){\frac {\partial f}{\partial \theta }}+\dots \\&\qquad \qquad \qquad \qquad \;\dots +(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2}){\frac {\partial ^{2}f}{\partial \lambda ^{2}}}+\lambda (2\lambda ^{2}-b^{2}-c^{2}){\frac {\partial f}{\partial \lambda }}{\Bigg \}}\end{aligned}}}

Kegelkoordinaten bieten sich bei der Lösung von Randwertaufgaben an, in denen die Ränder kugel- oder kegelförmig sind.^{[1] :1} Die Lösung wird erleichtert, wenn eine Trennung der Variablen gelingt, was in Kegelkoordinaten immer möglich ist^{[1] :7[2] :511} Dazu wird der Separationsansatz ^{[1] :39}

${\displaystyle \phi (r,\theta ,\lambda )=R(r)\cdot \Theta (\theta )\cdot \Lambda (\lambda )}${\displaystyle \phi (r,\theta ,\lambda )=R(r)\cdot \Theta (\theta )\cdot \Lambda (\lambda )}

in die Helmholtz-Gleichung ${\displaystyle \Delta \phi +\kappa ^{2}\phi =0}${\displaystyle \Delta \phi +\kappa ^{2}\phi =0} eingesetzt. Die Faktoren bestimmen sich dann aus den drei gewöhnlichen Differenzialgleichungen ^{[1] :39}

${\displaystyle {\begin{aligned}{\frac {\partial ^{2}R}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial R}{\partial r}}+\left(\kappa ^{2}-{\frac {\alpha _{2}}{r^{2}}}\right)R=&0\\(\theta ^{2}-b^{2})(c^{2}-\theta ^{2}){\frac {\partial ^{2}\Theta }{\partial \theta ^{2}}}-\theta (2\theta ^{2}-b^{2}-c^{2}){\frac {\partial \Theta }{\partial \theta }}+(\alpha _{2}\theta ^{2}-\alpha _{3})\Theta =&0\\(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2}){\frac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}+\lambda (2\lambda ^{2}-b^{2}-c^{2}){\frac {\partial \Lambda }{\partial \lambda }}-(\alpha _{2}\lambda ^{2}-\alpha _{3})\Lambda =&0\end{aligned}}}${\displaystyle {\begin{aligned}{\frac {\partial ^{2}R}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial R}{\partial r}}+\left(\kappa ^{2}-{\frac {\alpha _{2}}{r^{2}}}\right)R=&0\\(\theta ^{2}-b^{2})(c^{2}-\theta ^{2}){\frac {\partial ^{2}\Theta }{\partial \theta ^{2}}}-\theta (2\theta ^{2}-b^{2}-c^{2}){\frac {\partial \Theta }{\partial \theta }}+(\alpha _{2}\theta ^{2}-\alpha _{3})\Theta =&0\\(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2}){\frac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}+\lambda (2\lambda ^{2}-b^{2}-c^{2}){\frac {\partial \Lambda }{\partial \lambda }}-(\alpha _{2}\lambda ^{2}-\alpha _{3})\Lambda =&0\end{aligned}}}

Denn die Helmholtz-Gleichung lautet mit dem Ansatz:

${\displaystyle {\begin{aligned}\Delta \phi =&{\frac {\partial ^{2}R}{\partial r^{2}}}\Theta \Lambda +{\frac {2}{r}}{\frac {\partial R}{\partial r}}\Theta \Lambda +\dots \\&\dots +{\frac {1}{r^{2}(\theta ^{2}-\lambda ^{2})}}{\Bigg \{}(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})R{\frac {\partial ^{2}\Theta }{\partial \theta ^{2}}}\Lambda -\theta (2\theta ^{2}-b^{2}-c^{2})R{\frac {\partial \Theta }{\partial \theta }}\Lambda +\dots \\&\qquad \qquad \qquad \quad \dots +(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2})R\Theta {\frac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}+\lambda (2\lambda ^{2}-b^{2}-c^{2})R\Theta {\frac {\partial \Lambda }{\partial \lambda }}{\Bigg \}}+\dots \\&\dots +\kappa ^{2}R\Theta \Lambda =0\end{aligned}}}${\displaystyle {\begin{aligned}\Delta \phi =&{\frac {\partial ^{2}R}{\partial r^{2}}}\Theta \Lambda +{\frac {2}{r}}{\frac {\partial R}{\partial r}}\Theta \Lambda +\dots \\&\dots +{\frac {1}{r^{2}(\theta ^{2}-\lambda ^{2})}}{\Bigg \{}(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})R{\frac {\partial ^{2}\Theta }{\partial \theta ^{2}}}\Lambda -\theta (2\theta ^{2}-b^{2}-c^{2})R{\frac {\partial \Theta }{\partial \theta }}\Lambda +\dots \\&\qquad \qquad \qquad \quad \dots +(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2})R\Theta {\frac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}+\lambda (2\lambda ^{2}-b^{2}-c^{2})R\Theta {\frac {\partial \Lambda }{\partial \lambda }}{\Bigg \}}+\dots \\&\dots +\kappa ^{2}R\Theta \Lambda =0\end{aligned}}}

Multiplikation mit ${\displaystyle {\tfrac {r^{2}}{R\Theta \Lambda }}}${\displaystyle {\tfrac {r^{2}}{R\Theta \Lambda }}} liefert umgestellt:

${\displaystyle {\begin{aligned}&{\frac {r^{2}}{R}}{\frac {\partial ^{2}R}{\partial r^{2}}}+{\frac {2r}{R}}{\frac {\partial R}{\partial r}}+\kappa ^{2}r^{2}+\dots \\&\dots +{\frac {1}{\theta ^{2}-\lambda ^{2}}}{\Bigg \{}(\theta ^{2}-b^{2})(c^{2}-\theta ^{2}){\frac {\frac {\partial ^{2}\Theta }{\partial \theta ^{2}}}{\Theta }}-\theta (2\theta ^{2}-b^{2}-c^{2}){\frac {\frac {\partial \Theta }{\partial \theta }}{\Theta }}+\dots \\&\qquad \qquad \qquad \dots +(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2}){\frac {\frac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}{\Lambda }}+\lambda (2\lambda ^{2}-b^{2}-c^{2}){\frac {\frac {\partial \Lambda }{\partial \lambda }}{\Lambda }}{\Bigg \}}=0\end{aligned}}}${\displaystyle {\begin{aligned}&{\frac {r^{2}}{R}}{\frac {\partial ^{2}R}{\partial r^{2}}}+{\frac {2r}{R}}{\frac {\partial R}{\partial r}}+\kappa ^{2}r^{2}+\dots \\&\dots +{\frac {1}{\theta ^{2}-\lambda ^{2}}}{\Bigg \{}(\theta ^{2}-b^{2})(c^{2}-\theta ^{2}){\frac {\frac {\partial ^{2}\Theta }{\partial \theta ^{2}}}{\Theta }}-\theta (2\theta ^{2}-b^{2}-c^{2}){\frac {\frac {\partial \Theta }{\partial \theta }}{\Theta }}+\dots \\&\qquad \qquad \qquad \dots +(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2}){\frac {\frac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}{\Lambda }}+\lambda (2\lambda ^{2}-b^{2}-c^{2}){\frac {\frac {\partial \Lambda }{\partial \lambda }}{\Lambda }}{\Bigg \}}=0\end{aligned}}}

Weil nur die Terme in der ersten Zeile vom Radius r abhängen, ergänzen sie sich in Summe zu einer Konstanten α₂:

${\displaystyle {\frac {r^{2}}{R}}{\frac {\partial ^{2}R}{\partial r^{2}}}+{\frac {2r}{R}}{\frac {\partial R}{\partial r}}+\kappa ^{2}r^{2}=\alpha _{2}}${\displaystyle {\frac {r^{2}}{R}}{\frac {\partial ^{2}R}{\partial r^{2}}}+{\frac {2r}{R}}{\frac {\partial R}{\partial r}}+\kappa ^{2}r^{2}=\alpha _{2}}

Diese eingesetzt erlaubt auch θ und λ zu trennen:

${\displaystyle {\begin{aligned}&(\theta ^{2}-b^{2})(c^{2}-\theta ^{2}){\frac {\frac {\partial ^{2}\Theta }{\partial \theta ^{2}}}{\Theta }}-\theta (2\theta ^{2}-b^{2}-c^{2}){\frac {\frac {\partial \Theta }{\partial \theta }}{\Theta }}+\alpha _{2}\theta ^{2}=\dots \\&\qquad \qquad \qquad \;\dots =(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2}){\frac {\frac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}{\Lambda }}+\lambda (2\lambda ^{2}-b^{2}-c^{2}){\frac {\frac {\partial \Lambda }{\partial \lambda }}{\Lambda }}-\alpha _{2}\lambda ^{2}\end{aligned}}}${\displaystyle {\begin{aligned}&(\theta ^{2}-b^{2})(c^{2}-\theta ^{2}){\frac {\frac {\partial ^{2}\Theta }{\partial \theta ^{2}}}{\Theta }}-\theta (2\theta ^{2}-b^{2}-c^{2}){\frac {\frac {\partial \Theta }{\partial \theta }}{\Theta }}+\alpha _{2}\theta ^{2}=\dots \\&\qquad \qquad \qquad \;\dots =(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2}){\frac {\frac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}{\Lambda }}+\lambda (2\lambda ^{2}-b^{2}-c^{2}){\frac {\frac {\partial \Lambda }{\partial \lambda }}{\Lambda }}-\alpha _{2}\lambda ^{2}\end{aligned}}}

Weil die Summe auf der linken Seite nur von θ und die auf der rechten nur von λ abhängt, ergeben beide eine Konstante α₃, was auf die drei oben angegebenen gewöhnlichen Differenzialgleichungen zur Bestimmung der Faktoren R,Θ und Λ führt.

Die im Hauptartikel angegebene Methode zur Separation der Helmholtz-Gleichung führt mit der Stäckel-Matrix^{[1] :37}

${\displaystyle \mathbf {S} ={\begin{pmatrix}1&-{\frac {1}{r^{2}}}&0\0円&{\frac {\vartheta ^{2}}{(c^{2}-\vartheta ^{2})(\vartheta ^{2}-b^{2})}}&{\frac {-1}{(c^{2}-\vartheta ^{2})(\vartheta ^{2}-b^{2})}}\0円&{\frac {-\lambda ^{2}}{(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2})}}&{\frac {1}{(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2})}}\end{pmatrix}}}${\displaystyle \mathbf {S} ={\begin{pmatrix}1&-{\frac {1}{r^{2}}}&0\0円&{\frac {\vartheta ^{2}}{(c^{2}-\vartheta ^{2})(\vartheta ^{2}-b^{2})}}&{\frac {-1}{(c^{2}-\vartheta ^{2})(\vartheta ^{2}-b^{2})}}\0円&{\frac {-\lambda ^{2}}{(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2})}}&{\frac {1}{(b^{2}-\lambda ^{2})(c^{2}-\lambda ^{2})}}\end{pmatrix}}}

und den Funktionen

${\displaystyle f_{1}=r^{2},\;f_{2}={\sqrt {(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})}},\;f_{3}={\sqrt {(b^{2}-\lambda ^{2}),円(c^{2}-\lambda ^{2})}}}${\displaystyle f_{1}=r^{2},\;f_{2}={\sqrt {(\theta ^{2}-b^{2})(c^{2}-\theta ^{2})}},\;f_{3}={\sqrt {(b^{2}-\lambda ^{2}),円(c^{2}-\lambda ^{2})}}}

auf ein vergleichbares Ergebnis.

1. ↑ ^{a b c d e f g h i j} P. Moon, D.E. Spencer: Field Theory Handbook. Including Coordinate Systems, Differential Equations and Their Solutions. 2. Auflage. Springer Verlag, Berlin, Heidelberg, New York 1971, ISBN 3-540-02732-7. 
2. ↑ ^{a b c} P. M. Morse, H. Feshbach: Methods of Theoretical Physics, Part I. McGraw-Hill, New York 1953 (archive.org). 
3. ↑ Eric Weisstein: Conical Coordinates. MathWorld, 22. Juni 2024, abgerufen am 23. Juni 2024 (englisch). 

Abgerufen von „https://de.wikipedia.org/w/index.php?title=Kegelkoordinaten&oldid=246236764"

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• Koordinatensystem

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