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}}} det ( g i j ) = r 4 sin 2 θ . {\displaystyle \det(g_{ij})=r^{4}\sin ^{2}\theta .\ } Yoy uzunligi differensialining kvadrati: 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 .} Kristoffel belgilari { r , θ , φ } {\displaystyle \{r,\;\theta ,\;\varphi \}} : 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 birlik vektorlar 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
Eyler burchaklari Gipersferik koordinatalar Manbalar
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