Sferik koordinatalar

Sferik koordinatalar sistemasi — uch oʻlchamli koordinatalar sistemasi boʻlib, fazodagi nuqtaning vaziyati uchta kattalik bilan (${\displaystyle r,\theta ,\varphi }$) bilan aniqlanadi. Bu yerda ${\displaystyle {\displaystyle r}}$ — koordinatalar boshigacha boʻlgan masofa, ${\displaystyle {\displaystyle \theta }}$ va ${\displaystyle {\displaystyle \varphi }}$— 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 ${\displaystyle (r,\;\theta ,\;\varphi )}$  berilgan boʻlsa, dekart koordinatalariga oʻtish uchun quyidagi formulalardan foydalaniladi:

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

Dekart koordinatalaridan sferik koordinatalarga oʻtish uchun esa:

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

Sferik koordinatalarga oʻtish yakobiani:

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

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

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

Silindrik koordinatalar sistemasi

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

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

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

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

Silindrik koordinatalardan sferik koordinatalarga oʻtish yakobiani :

${\displaystyle J=r}$ 

Sferik koordinatalar sistemasida differensiallash va integrallash

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

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

bu yerda

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

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

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

Sferik koordinatalar ortogonal hisoblanadi. Shu sababli ularning metrik tenzori diagonal koʻrinishda boʻladi:

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

• ${\displaystyle \det(g_{ij})=r^{4}\sin ^{2}\theta .\ }$ 
• Yoy uzunligi differensialining kvadrati:

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

• Lame koeffitsiyentlari:

${\displaystyle H_{r}=1,\quad H_{\theta }=r,\quad H_{\varphi }=r\sin \theta .}$ 

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

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

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

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

 

Sferik koordinatalar sistemasida birlik vektorlar

Sferik koordinatalar sistemasida masofa

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

${\displaystyle {\begin{aligned}{\mathbf {r} }&=(r,\theta ,\varphi ),\\{\mathbf {r} '}&=(r',\theta ',\varphi ')\end{aligned}}}$ 

U holda ushbu nuqtalar orasidagi masofani quyidagi formula orqali hisoblash mumkin:

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

Harakat tenglamasi

Nuqtaning vaziyati sferik koordinatalarda quyidagi koʻrinishda berilgan boʻlsin:

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

U holda uning tezligi:

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

hamda tezlanishi:

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

ga teng boʻladi.

Burchak momenti:

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

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

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

Yana qarang

• Eyler burchaklari
• Gipersferik koordinatalar