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.

## 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:

: \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}
* \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 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
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* Python tilida yozilgan programma