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==Einschub: Transformation auf Kugelkoordinaten== :<math>\begin{align} & \left( r,\vartheta ,\phi \right)=\left( {{q}_{1}},{{q}_{2}},{{q}_{3}} \right) \\ & x=r\cos \phi \sin \vartheta \\ & y=r\sin \phi \sin \vartheta \\ & z=r\cos \vartheta \\ \end{align}</math> :<math>\begin{align} & {{{\vec{v}}}_{{}}}=\sum\limits_{j}{{}}\left( \frac{\partial {{{\vec{r}}}_{{}}}}{\partial {{q}_{j}}} \right){{{\dot{q}}}_{j}} \\ & \\ \end{align}</math> In Komponenten ergibt sich somit: :<math>\begin{align} & {{v}_{x}}=\frac{dx}{dt}=\frac{\partial x}{\partial r}\dot{r}+\frac{\partial x}{\partial \vartheta }\dot{\vartheta }+\frac{\partial x}{\partial \phi }\dot{\phi }=\sin \vartheta \cos \phi \dot{r}+r\cos \vartheta \cos \phi \dot{\vartheta }-r\sin \vartheta \sin \phi \dot{\phi } \\ & {{v}_{y}}=\frac{dy}{dt}=\frac{\partial y}{\partial r}\dot{r}+\frac{\partial y}{\partial \vartheta }\dot{\vartheta }+\frac{\partial y}{\partial \phi }\dot{\phi }=\sin \vartheta \sin \phi \dot{r}+r\cos \vartheta \sin \phi \dot{\vartheta }+r\sin \vartheta \cos \phi \dot{\phi } \\ & {{v}_{z}}=\frac{dz}{dt}=\frac{\partial z}{\partial r}\dot{r}+\frac{\partial z}{\partial \vartheta }\dot{\vartheta }+\frac{\partial z}{\partial \phi }\dot{\phi }=\cos \vartheta \dot{r}-r\sin \vartheta \dot{\vartheta } \\ & \\ \end{align}</math> Es läßt sich eine Funktionalmatrix zusammenstellen: :<math>\left( \begin{matrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \vartheta } & \frac{\partial x}{\partial \phi } \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \vartheta } & \frac{\partial y}{\partial \phi } \\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial \vartheta } & \frac{\partial z}{\partial \phi } \\ \end{matrix} \right)=\left( \begin{matrix} \sin \vartheta \cos \phi & r\cos \vartheta \cos \phi & -r\sin \vartheta \sin \phi \\ in\vartheta \sin \phi & r\cos \vartheta \sin \phi & r\sin \vartheta \cos \phi \\ \cos \vartheta & -r\sin \vartheta & 0 \\ \end{matrix} \right)</math> :<math>\begin{align} & T=\frac{1}{2}\sum\limits_{j,k}{{{T}_{jk}}}{{{\dot{q}}}_{j}}{{{\dot{q}}}_{k}} \\ & {{T}_{jk}}={{T}_{kj}}\approx \sum\limits_{i}{{{m}_{i}}{{\left( \frac{\partial {{{\vec{r}}}_{i}}}{\partial {{q}_{j}}} \right)}_{0}}{{\left( \frac{\partial {{{\vec{r}}}_{i}}}{\partial {{q}_{j}}} \right)}_{0}}} \\ & {{T}_{jk}}=m\left[ \left( \frac{\partial x}{\partial {{q}_{j}}} \right)\left( \frac{\partial x}{\partial {{q}_{k}}} \right)+\left( \frac{\partial y}{\partial {{q}_{j}}} \right)\left( \frac{\partial y}{\partial {{q}_{k}}} \right)+\left( \frac{\partial z}{\partial {{q}_{j}}} \right)\left( \frac{\partial z}{\partial {{q}_{k}}} \right) \right] \\ \end{align}</math> :<math>\begin{align} & {{T}_{11}}=m\left( {{\sin }^{2}}\vartheta {{\cos }^{2}}\phi +{{\sin }^{2}}\vartheta {{\sin }^{2}}\phi +{{\cos }^{2}}\vartheta \right)=m \\ & {{T}_{22}}=m{{r}^{2}}\left( {{\cos }^{2}}\vartheta {{\cos }^{2}}\phi +{{\cos }^{2}}\vartheta {{\sin }^{2}}\phi +{{\sin }^{2}}\vartheta \right)=m{{r}^{2}} \\ & {{T}_{33}}=m{{r}^{2}}({{\sin }^{2}}\vartheta {{\sin }^{2}}\phi +{{\sin }^{2}}\vartheta {{\cos }^{2}}\phi )=m{{r}^{2}}{{\sin }^{2}}\vartheta \\ \end{align}</math> Diese Wert hängen dabei von den gewählten Koordinaten, also den qj ab. Aus diesem Grund (um dies zu erreichen) wurden ja gerade die qj so eingeführt. :<math>\begin{align} & {{T}_{12}}={{T}_{21}}=mr\left( \sin \vartheta \cos \phi \cos \vartheta \cos \phi +\sin \vartheta \sin \phi \cos \vartheta \sin \phi -\sin \vartheta \cos \vartheta \right)=0 \\ & {{T}_{13}}={{T}_{31}}=0 \\ & {{T}_{23}}={{T}_{32}}=0 \\ \end{align}</math> :<math>\begin{align} & {{T}_{jk}}=\left( \begin{matrix} m & 0 & 0 \\ 0 & m{{r}^{2}} & 0 \\ 0 & 0 & m{{r}^{2}}{{\sin }^{2}}\vartheta \\ \end{matrix} \right) \\ & T=\frac{1}{2}m\left( {{{\dot{r}}}^{2}}+{{r}^{2}}{{{\dot{\vartheta }}}^{2}}+{{r}^{2}}{{\sin }^{2}}\vartheta {{{\dot{\phi }}}^{2}} \right) \\ \end{align}</math>
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