Sferik koordinatalar

Sferik koordinatalar sistemasi — uch oʻlchamli koordinatalar sistemasi boʻlib, fazodagi nuqtaning vaziyati uchta kattalik bilan ($r, \theta, \varphi$) bilan aniqlanadi. Bu yerda ${\displaystyle r}$ — koordinatalar boshigacha boʻlgan masofa, ${\displaystyle \theta }$ va ${\displaystyle \varphi }$— mos holda zenit va azimutal burchaklar.

Zenit va azimut tushunchalari astronomiyada keng qoʻllanadi. Zenit — ixtiyoriy tanlangan nuqta (kuzatish nuqtasi) dan vertikal yuqoriga yoʻnalgan boʻlib, fundamental tekislikda yotadi. Astronomiyada fundamental tekislik sifatida ekvator yotgan tekislik yoki ekliptika tekisligi olinadi. Azimut — fundamental tekislikdagi ixtiyoriy tanlangan nur bilan boshlangʻich kuzatish nuqtasi orasidagi burchak.

1-rasm. Ixtiyoriy nuqtaning fazodagi vaziyatini uchta dekart koordinatalari yoki uchta sferik koordinatalar orqali aniqlash mumkin.

Boshqa koordinata sistemalariga oʻtish

Dekart koordinatalar sistemasi

Agar nuqtaning sferik koordinatalari $(r,\;\theta,\;\varphi)$ berilgan boʻlsa, dekart koordinatalariga oʻtish uchun quyidagi formulalardan foydalaniladi:

$\begin{cases} x=r\sin\theta\cos\varphi, \\ y=r\sin\theta\sin\varphi, \\ z=r\cos\theta. \end{cases}$

Dekart koordinatalaridan sferik koordinatalarga oʻtish uchun esa:

$\begin{cases} r=\sqrt{x^2+y^2+z^2}, \\ \theta=\arccos\dfrac{z}{\sqrt{x^2+y^2+z^2}}=\mathrm{arctg}\dfrac{\sqrt{x^2+y^2}}{z}, \\ \varphi=\mathrm{arctg}\dfrac{y}{x}. \end{cases}$

Sferik koordinatalarga oʻtish yakobiani:

Failed to parse (Unknown environment alignat): {\displaystyle \begin{alignat}{2}

J & =\frac{\partial(x,y,z)}{\partial (r,\theta,\varphi)}=\begin{vmatrix} \sin \theta \cos \varphi & r \cos \theta \cos \varphi & -r\sin\theta\sin\varphi\\ \sin \theta \sin \varphi & r \cos \theta \sin \varphi & r \sin \theta \cos \varphi \\ \cos \theta & - r\sin\theta & 0 \end{vmatrix} = \\ & =\cos \theta (r^2 \cos \varphi^2 \cos \theta \sin \theta + r^2\sin^2\varphi \cos\theta\sin\theta) + r\sin\theta(r\sin^2\theta\cos^2\varphi + r\sin^2\theta\sin^2\varphi) = \\ & = r^2 \cos^2\theta \sin \theta + r^2 \sin^2 \theta \sin\theta =\\ &= r^2\sin\theta. \end{alignat}}

Shunday qilib, dekart koordinatalaridan sferik koordinatalarga oʻtishdagi hajm elementi quyidagi koʻrinishga ega boʻladi:

$\mathrm{d}V = \mathrm{d}x\, \mathrm{d}y\, \mathrm{d}z = J(r,\theta,\varphi) \,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\varphi = r^2 \sin \theta \, \,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\varphi$

Silindrik koordinatalar sistemasi

Agar nuqtaning silindrik koordinatalari berilgan boʻlsa, sferik koordinatalarga oʻtish uchun quyidagi formulalardan foydalaniladi:

$\begin{cases} \rho = r\sin\theta\\ \varphi = \varphi\\ z = r\cos\theta \end{cases}$

Yoki aksincha, sferik koordinatalardan silindrik koordinatalarga oʻtish uchun quyidagi formulalardan foydalaniladi:

$\begin{cases} r=\sqrt{\rho^2+z^2}, \\ \theta=\mathrm{arctg}\dfrac{\rho}{z}, \\ \varphi=\varphi. \end{cases}$

Silindrik koordinatalardan sferik koordinatalarga oʻtish yakobiani :

$J = r$

Sferik koordinatalar sistemasida differensiallash va integrallash

$(r,\theta,\varphi)$ nuqtadan $(r+\mathrm{d}r, \,\theta+\mathrm{d}\theta, \, \varphi+\mathrm{d}\varphi)$ nuqtaga oʻtkazilgan vektor $\mathrm{d}\mathbf{r}$ ning uzunligi quyidagiga teng:

$\mathrm{d}\mathbf{r} = \mathrm{d}r\,\boldsymbol{\hat r} + r\,\mathrm{d}\theta \,\boldsymbol{\hat\theta } + r \sin{\theta} \, \mathrm{d}\varphi\,\mathbf{\boldsymbol{\hat \varphi}},$

bu yerda

$\boldsymbol{\hat r} =\sin \theta \cos \varphi \boldsymbol{\hat{\imath}} + \sin \theta \sin \varphi \boldsymbol{\hat{\jmath}} + \cos \theta \boldsymbol{\hat{k}}$

$\boldsymbol{\hat\theta } =\cos \theta \cos \varphi \boldsymbol{\hat{\imath}} + \cos \theta \sin \varphi \boldsymbol{\hat{\jmath}} -\sin \theta \boldsymbol{\hat{k}}$

$\boldsymbol{\hat \varphi} =-\sin \varphi \boldsymbol{\hat{\imath}} + \cos \varphi \boldsymbol{\hat{\jmath}}$ Sferik koordinatalar ortogonal hisoblanadi. Shu sababli ularning metrik tenzori diagonal koʻrinishda boʻladi:

$g_{ij}=\begin{pmatrix} 1 & 0 & 0\\ 0 & r^2 & 0\\ 0 & 0 & r^2\sin^2\theta \end{pmatrix},\quad g^{ij}=\begin{pmatrix} 1 & 0 & 0\\ 0 & \dfrac{1}{r^2} & 0\\ 0 & 0 & \dfrac{1}{r^2\sin^2\theta} \end{pmatrix}$

• Failed to parse (trailing backslash): {\displaystyle \det(g_{ij})=r^4\sin^2\theta.\}
• Yoy uzunligi differensialining kvadrati:

$ds^2=dr^2+r^2\,d\theta^2+r^2\sin^2\theta\,d\varphi^2.$

• Lame koeffitsiyentlari:

$H_r=1,\quad H_\theta=r,\quad H_\varphi=r\sin\theta.$

• Kristoffel belgilari $\{r,\;\theta,\;\varphi\}$:

$\Gamma^1_{22}=-r,\quad \Gamma^1_{33}=-r\sin^2\theta,$

$\Gamma^2_{21}=\Gamma^2_{12}=\Gamma^3_{13}=\Gamma^3_{31}=\frac{1}{r},$

$\Gamma^2_{33}=-\cos\theta\sin\theta,\quad \Gamma^3_{23}=\Gamma^3_{32}=\mathrm{ctg}\,\theta.$

Sferik koordinatalar sistemasida birlik vektorlar

Sferik koordinatalar sistemasida masofa

Fazodagi vaziyati sferik koordinatalar sistemasida berilgan ikki nuqtaning joylashuvi quyidagicha boʻlsin:

$\begin{align$

{\mathbf r} &= (r,\theta,\varphi), \\
{\mathbf r'} &= (r',\theta',\varphi')

\end{align}} U holda ushbu nuqtalar orasidagi masofani quyidagi formula orqali hisoblash mumkin:

$\begin{align$

{\mathbf D} &= \sqrt{r^2+r'^2-2rr'(\sin{\theta}\sin{\theta'}\cos{(\varphi-\varphi')} + \cos{\theta}\cos{\theta'})}

\end{align}}

Harakat tenglamasi

Nuqtaning vaziyati sferik koordinatalarda quyidagi koʻrinishda berilgan boʻlsin:

$\mathbf{r} = r \mathbf{\hat r}.$

U holda uning tezligi:

$\mathbf{v} = \dot{r} \mathbf{\hat r} + r\,\dot\theta\,\hat{\boldsymbol\theta } + r\,\dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\varphi}},$

hamda tezlanishi:

$\begin{align} \mathbf{a} = {} & \left( \ddot{r} - r\,\dot\theta^2 - r\,\dot\varphi^2\sin^2\theta \right)\mathbf{\hat r} \\ & {} + \left( r\,\ddot\theta + 2\dot{r}\,\dot\theta - r\,\dot\varphi^2\sin\theta\cos\theta \right) \hat{\boldsymbol\theta } \\ & {} + \left( r\ddot\varphi\,\sin\theta + 2\dot{r}\,\dot\varphi\,\sin\theta + 2 r\,\dot\theta\,\dot\varphi\,\cos\theta \right) \hat{\boldsymbol\varphi}. \end{align}$

ga teng boʻladi.

Burchak momenti:

$\mathbf{L} = m \mathbf{r} \times \mathbf{v} = m r^2 (\dot\theta\,\hat{\boldsymbol\varphi } - \dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\theta}} ).$

$\varphi$ oʻzgarmas boʻlganda yoki $\theta=\frac{\pi}{2}$ boʻlganda, moddiy nuqtaning harakat tenglamasi qutb koordinatalar sistemasiga oʻtadi.

$\mathbf{L }= -i\hbar ~\mathbf{r} \times \nabla =i \hbar \left(\frac{\hat{\boldsymbol{\theta}}}{\sin(\theta)} \frac{\partial}{\partial\phi} - \hat{\boldsymbol{\phi}} \frac{\partial}{\partial\theta}\right) .$

Yana qarang

• Eyler burchaklari
• Gipersferik koordinatalar

Manbalar

• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, 1953 — 658-bet. ISBN 0-07-043316-X. 
• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand, 1956 — 177–178-bet. 
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, 1961 — 174–175-bet. ASIN B0000CKZX7. 
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag, 1967 — 95–96-bet. 
• Moon P, Spencer DE „Spherical Coordinates (r, θ, ψ)“, . Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, corrected 2nd ed., 3rd print, New York: Springer-Verlag, 1988 — 24–27 (Table 1.05)-bet. ISBN 978-0-387-18430-2. 
• Duffett-Smith P, Zwart J. Practical Astronomy with your Calculator or Spreadsheet, 4th Edition. New York: Cambridge University Press, 2011 — 34-bet. ISBN 978-0521146548. 
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