Sferik Koordinatalar

---

Sferik koordinatalar sistemasi  — uch oʻlchamli koordinatalar sistemasi boʻlib, fazodagi nuqtaning vaziyati uchta kattalik bilan (



r
,
θ
,
φ


{\displaystyle r,\theta ,\varphi }

) bilan aniqlanadi. Bu yerda





r




{\displaystyle {\displaystyle r}}

 — koordinatalar boshigacha boʻlgan masofa,





θ




{\displaystyle {\displaystyle \theta }}

va





φ




{\displaystyle {\displaystyle \varphi }}

— mos holda zenit va azimutal burchaklar.

Zenit va azimut tushunchalari astronomiyada keng qoʻllaniladi. 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
,

θ
,

φ
)


{\displaystyle (r,\;\theta ,\;\varphi )}

berilgan boʻlsa, dekart koordinatalariga oʻtish uchun quyidagi formulalardan foydalaniladi:






{



x
=
r
sin
⁡
θ
cos
⁡
φ
,




y
=
r
sin
⁡
θ
sin
⁡
φ
,




z
=
r
cos
⁡
θ
.








{\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:






{



r
=



x

2


+

y

2


+

z

2




,




θ
=
arccos
⁡



z


x

2


+

y

2


+

z

2






=

a
r
c
t
g






x

2


+

y

2



z



,




φ
=

a
r
c
t
g




y
x



.








{\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:








J



=



∂
(
x
,
y
,
z
)


∂
(
r
,
θ
,
φ
)



=


|



sin
⁡
θ
cos
⁡
φ


r
cos
⁡
θ
cos
⁡
φ


−
r
sin
⁡
θ
sin
⁡
φ




sin
⁡
θ
sin
⁡
φ


r
cos
⁡
θ
sin
⁡
φ


r
sin
⁡
θ
cos
⁡
φ




cos
⁡
θ


−
r
sin
⁡
θ


0



|


=






=
cos
⁡
θ
(

r

2


cos
⁡

φ

2


cos
⁡
θ
sin
⁡
θ
+

r

2



sin

2


⁡
φ
cos
⁡
θ
sin
⁡
θ
)
+
r
sin
⁡
θ
(
r

sin

2


⁡
θ

cos

2


⁡
φ
+
r

sin

2


⁡
θ

sin

2


⁡
φ
)
=






=

r

2



cos

2


⁡
θ
sin
⁡
θ
+

r

2



sin

2


⁡
θ
sin
⁡
θ
=






=

r

2


sin
⁡
θ
.






{\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:





d

V
=

d

x


d

y


d

z
=
J
(
r
,
θ
,
φ
)


d

r


d

θ


d

φ
=

r

2


sin
⁡
θ



d

r


d

θ


d

φ


{\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:






{



ρ
=
r
sin
⁡
θ




φ
=
φ




z
=
r
cos
⁡
θ








{\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:






{



r
=



ρ

2


+

z

2




,




θ
=

a
r
c
t
g




ρ
z



,




φ
=
φ
.








{\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 :




J
=
r


{\displaystyle J=r}

Sferik koordinatalar sistemasida differensiallash va integrallash

---

(
r
,
θ
,
φ
)


{\displaystyle (r,\theta ,\varphi )}

nuqtadan



(
r
+

d

r
,

θ
+

d

θ
,

φ
+

d

φ
)


{\displaystyle (r+\mathrm {d} r,\,\theta +\mathrm {d} \theta ,\,\varphi +\mathrm {d} \varphi )}

nuqtaga oʻtkazilgan vektor




d


r



{\displaystyle \mathrm {d} \mathbf {r} }

ning uzunligi quyidagiga teng:





d


r

=

d

r




r
^



+
r


d

θ




θ
^



+
r
sin
⁡

θ



d

φ




φ
^



,


{\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







r
^



=
sin
⁡
θ
cos
⁡
φ



ı
^



+
sin
⁡
θ
sin
⁡
φ



ȷ
^



+
cos
⁡
θ



k
^





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








θ
^



=
cos
⁡
θ
cos
⁡
φ



ı
^



+
cos
⁡
θ
sin
⁡
φ



ȷ
^



−
sin
⁡
θ



k
^





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








φ
^



=
−
sin
⁡
φ



ı
^



+
cos
⁡
φ



ȷ
^





{\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:





g

i
j


=


(



1


0


0




0



r

2




0




0


0



r

2



sin

2


⁡
θ



)


,


g

i
j


=


(



1


0


0




0





1

r

2







0




0


0





1


r

2



sin

2


⁡
θ







)




{\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}}}





d

s

2


=
d

r

2


+

r

2



d

θ

2


+

r

2



sin

2


⁡
θ

d

φ

2


.


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






H

r


=
1
,


H

θ


=
r
,


H

φ


=
r
sin
⁡
θ
.


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






Γ

22


1


=
−
r
,


Γ

33


1


=
−
r

sin

2


⁡
θ
,


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






Γ

21


2


=

Γ

12


2


=

Γ

13


3


=

Γ

31


3


=


1
r


,


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






Γ

33


2


=
−
cos
⁡
θ
sin
⁡
θ
,


Γ

23


3


=

Γ

32


3


=

c
t
g


θ
.


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

Sferik koordinatalar sistemasida masofa

---

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










r





=
(
r
,
θ
,
φ
)
,







r

′





=
(

r
′

,

θ
′

,

φ
′

)






{\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:










D





=



r

2


+

r

′

2



−
2
r

r
′

(
sin
⁡

θ

sin
⁡


θ
′


cos
⁡

(
φ
−

φ
′

)

+
cos
⁡

θ

cos
⁡


θ
′


)








{\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:





r

=
r



r
^



.


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


U holda uning tezligi:





v

=



r
˙






r
^



+
r




θ
˙







θ
^



+
r




φ
˙



sin
⁡
θ




φ
^



,


{\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:









a

=






(




r
¨



−
r





θ
˙




2


−
r





φ
˙




2



sin

2


⁡
θ

)




r
^












+

(

r




θ
¨



+
2



r
˙







θ
˙



−
r





φ
˙




2


sin
⁡
θ
cos
⁡
θ

)




θ
^












+

(

r



φ
¨




sin
⁡
θ
+
2



r
˙







φ
˙




sin
⁡
θ
+
2
r




θ
˙







φ
˙




cos
⁡
θ

)




φ
^



.






{\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:





L

=
m

r

×

v

=
m

r

2


(



θ
˙







φ
^



−



φ
˙



sin
⁡
θ




θ
^



)
.


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





φ


{\displaystyle \varphi }

oʻzgarmas boʻlganda yoki



θ
=


π
2




{\displaystyle \theta ={\frac {\pi }{2}}}

boʻlganda, moddiy nuqtaning harakat tenglamasi qutb koordinatalar sistemasiga oʻtadi.





L

=
−
i
ℏ
 

r

×
∇
=
i
ℏ

(





θ
^



sin
⁡
(
θ
)





∂

∂
ϕ



−



ϕ
^





∂

∂
θ




)

.


{\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

---

Manbalar

---

uz.wikipedia.org