Fazoda to‘g‘ri burchakli koordinatalar sistemasi

Sferik koordinatalar sistemasida masofaTahrirlash

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 tenglamasiTahrirlash

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).}

Shuningdek qarangTahrirlash

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  Eyler burchaklari
  
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  Gipersferik koordinatalar