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und $V = abc$ folgt $a = b = c = \sqrt[3]{V}$. + + Überprüfung ob ein Minimum vorliegt: + \begin{align*} + H E(a,b,c) = \frac{h^2}{4m} \begin{pmatrix} \frac{3}{a^{4}} & 0 & 0 \\ + 0 & \frac{3}{b^{4}} & 0 \\ + 0 & 0 & \frac{3}{c^{4}} + \end{pmatrix} + .\end{align*} + $H E(a,b,c)$ nach Hauptminorenkriterium positiv definit, damit liegt bei $a = b = c = \sqrt[3]{V}$ + ein Minimum vor. +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Für die Zwangsbedingung gilt + \[ + \tan \alpha = \frac{z}{x - \xi(t)} \implies x \sin \alpha - \xi(t) \sin \alpha - z \cos \alpha = 0. \qquad (*) + \] + \item Damit folgen die Langrange-Gleichungen 1. Art: + \[ + \begin{pmatrix} 0 \\ 0 \\ -mg \end{pmatrix} - m\ddot{\vec{x}} + \lambda \begin{pmatrix} \sin \alpha \\ 0 \\ -\cos\alpha \end{pmatrix} = 0 \implies + \begin{cases} + -m\ddot{x} + \lambda \sin \alpha = 0 \implies \lambda = \frac{m}{\sin\alpha} \ddot{x} \\ + m\ddot{y} = 0 \qquad \stackrel{\vec{x}(0) = \dot{\vec{x}}(0) = 0} \implies \qquad y = 0\\ + mg + m\ddot{z} + \lamdba \cos\alpha = 0 + \end{cases} + .\] Mit $(*)$ folgt + \[ + x = \xi + z \frac{\cos\alpha}{\sin\alpha} \implies \ddot{x} + = \ddot{\xi} + \ddot{z} \frac{\cos\alpha}{\sin\alpha} + .\] Damit folgt für $z$: + \begin{align*} + &g + \ddot{z} + + \frac{\cos\alpha}{\sin\alpha}\left( \ddot{\xi} + \ddot{z} \frac{\cos\alpha}{\sin\alpha}\right) + = 0 \\ + \implies + &\ddot{z} = - g \sin^2\alpha - \sin\alpha \cos\alpha \ddot{\xi} + \intertext{Mit $\vec{x}(0) = 0$, $\dot{\vec{x}}(0) = 0$, $\xi(0) = 0$ und $\dot{\xi}(0) = 0$ + folgt} + &z = -\frac{1}{2}g\sin^2\alpha \cdot t^2 - \frac{1}{2}\sin(2\alpha)\xi + \intertext{Eingesetzt in $(*)$ folgt für $x$} + &x = \sin^2\alpha \cdot \xi - \frac{1}{4} g \sin(2 \alpha) t^2 + \intertext{Damit folgt insgesamt} + &\vec{x} = \begin{pmatrix} \sin^2\alpha \cdot \xi - \frac{1}{4} g \sin(2 \alpha) t^2 \\ + 0 \\ + -\frac{1}{2}g\sin^2\alpha \cdot t^2 - \frac{1}{2}\sin(2\alpha)\xi + \end{pmatrix} + .\end{align*} + \item Die Zwangskraft ist gegeben durch + \begin{align*} + \vec{Z} &= \lambda \vec{\nabla} f = \lambda \begin{pmatrix} \sin\alpha \\ 0 \\ -\cos\alpha \end{pmatrix} + \intertext{Mit $\lambda = \frac{m}{\sin\alpha}\ddot{x}$ folgt direkt} + \vec{Z} &= \begin{pmatrix} m \ddot{\xi} \sin^2\alpha - \frac{1}{2} mg \sin(2\alpha) \\ + 0 \\ + - \frac{m}{2} \ddot{\xi} \sin(2\alpha) + mg \cos^2\alpha + \end{pmatrix} + .\end{align*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es liege ein Potential mit $\vec{F} = - \vec{\nabla} V(\vec{x})$ vor. Damit gilt + Energieerhaltung und es folgt + \begin{align*} + &\frac{m}{2} \dot{\vec{x}}^2 + V(\vec{x}) = E \\ + \implies & \left|\frac{\text{d}\vec{x}}{\d t}\right| = \sqrt{\frac{2}{m}(E - V(\vec{x}))} + \intertext{Durch Trennung der Variablen folgt} + &\d t = \frac{|\text{d}\vec{x}|}{\sqrt{\frac{2}{m} (E - V(\vec{x}))}} + \implies \Delta t = \int_{\vec{x}_0}^{\vec{x}_E} + \frac{|\text{d}\vec{x}|}{\sqrt{\frac{2}{m} (E - V(\vec{x}))}} + \intertext{Mit $\vec{x} = \begin{pmatrix} x \\ -f(x) \end{pmatrix} $ folgt} + & \frac{\text{d}\vec{x}}{\d x} = \begin{pmatrix} 1 \\ -f'(x) \end{pmatrix} + \implies \left| \frac{\text{d}\vec{x}}{\d x}\right| = \sqrt{1 + f'(x)^2} + \intertext{Zusammen folgt} + &\Delta t = \int_{x_0}^{x_E} \frac{\sqrt{1 + f'(x)^2}}{\sqrt{\frac{2}{m} (E - V(\vec{x}))}} \d x + .\end{align*} + \item Mit $E = 0$, $x_0 = 0$, $x_E = 1$ und $f(x) = x$ folgt + \[ + \Delta t = \int_{0}^{1} \frac{\sqrt{2} }{\sqrt{2 g} } \d x = \frac{2}{\sqrt{g} } \sqrt{x} + \Big|_{0}^{1} = \frac{2}{\sqrt{g} } + .\] + \end{enumerate} \end{aufgabe} \end{document}