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+ \intertext{Damit folgt für die verallgemeinerten Impulse} + \frac{\partial L}{\partial \dot{\rho}} &= m \dot{\rho} =: p_{\rho} \\ + \frac{\partial L}{\partial \dot{\varphi}} &= m \rho^2 \dot{\varphi} := p_{\varphi} \\ + \frac{\partial L}{\partial \dot{z}} &= m \dot{z} =: p_z + \intertext{Damit folgt die Hamiltonfunktion} + H(\vec{p}, \vec{q}) &= \frac{p_{\rho}^2}{m} + \frac{p_{\varphi}^2}{m \rho^2} + \frac{p_z^2}{m} + - \frac{m}{2} \left( \frac{p_{\rho}^2}{m^2} + \rho^2 \frac{p_{\varphi}^2}{m^2 \rho^{4}} + + \frac{p_{z}^2}{m^2}\right) + V(\rho)\\ + &= \frac{2}{m} \left( p_{\rho}^2 + \frac{p_{\varphi}^2}{\rho^2} + p_{z}^2 \right) + V(\rho) + \intertext{Als Erhaltungsgrößen folgen damit sofort} + \frac{\partial L}{\partial \varphi} &= 0 \implies \frac{\d}{\d t} p_{\varphi} = 0 \\ + \frac{\partial L}{\partial z} &= 0 \implies \frac{\d}{\d t} p_{z} = 0 \\ + \frac{\partial H}{\partial t} &= 0 \implies \frac{\d}{\d t} H = 0 \implies E = \text{konst} + .\end{align*} +\end{aufgabe} + +\begin{aufgabe}[Brachistochrone] + \begin{align*} + T[f] = \frac{1}{\sqrt{2g} } \int_{x_0}^{x_E} \sqrt{\frac{1 + [f'(x)]^2}{f(x)}} \d x + .\end{align*} + \begin{enumerate}[a)] + \item Die Lagrange Funktion und der verallgemeinerte Impuls sind damit gegeben als + \begin{align*} + L &= \frac{\sqrt{1 + f'^2} }{\sqrt{2gf} } \\ + p &= \frac{\partial L}{\partial f'} = \frac{f'}{\sqrt{2 g f( 1+f'^2)}} + \intertext{Damit folgt für die Hamilton-Funktion, ausgedrückt durch $f'$ statt $p$} + H(f, f') &= \frac{f'^2}{\sqrt{2 g f(1+f'^2)} } - \frac{\sqrt{1 + f'^2} }{\sqrt{2gf} } \\ + &= - \frac{1}{\sqrt{2 g f (1 + f'^2)}} + \intertext{Wegen $\frac{\partial H}{\partial t}$, folgt für die Konstante $E > 0$} + E &:= - H = \frac{1}{\sqrt{2 g f(1+f'^2)} } + .\end{align*} + Damit folgt als DGL + \[ + E \sqrt{2 g f (1+ f'^2)} = 1 + .\] + \item Mit $f(\varphi) = \frac{1 - \cos\varphi}{4 g E^2}$ und + $ x(\varphi) = \frac{\varphi - \sin\varphi}{4 g E^2}$ folgt + \begin{align*} + \frac{\d f}{\d x} = \frac{\d f}{\d \varphi} \frac{\d \varphi}{\d x} + = \frac{\sin \varphi}{1 - \cos \varphi} + .\end{align*} + Damit folgt + \[ + E \sqrt{2g \frac{1 - \cos \varphi}{4gE^2} \left( 1 + \frac{\sin^2\varphi}{(1 - \cos\varphi)^2} + \right) } + = \sqrt{\frac{1 - \cos\varphi}{2} + \frac{1 + \cos\varphi}{2}} = 1 + .\] + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe}[Verständnisfragen] + \begin{enumerate}[a)] + \item Koordinaten von denen die Lagrange Funktion nicht explizit abhängt, + heißen zyklisch. Dann ist der kanonisch konjugierte Impuls zeitlich konstant. + Ihr Nullpunkt kann beliebig verschoben werden, ohne die Bewegungsgleichungen + zu ändern: + \[ + q_i \to q_i + c + .\] + \item Das Hamilton'sche Prinzip besagt, dass die Wirkung entlang der wirklichen + Bahn eines Massenpunkts zwischen zwei Punkten extremal wird. Die Wirkung + ist das Zeitintegral über die Lagrange Funktion: + \[ + \delta S[q(t)] = \delta \left[\int_{t_0}^{t_1} L(q, \dot{q}, t) \d t \right] = 0 + .\] + \item Nein. Die Lagrange-Funktion kann um die Zeitableitung einer beliebigen Funktion $f(q, t)$ + ergänzt werden + \[ + L \to L + \frac{\d f(q, t)}{\d t} + ,\] denn dadurch ändert sich die Wirkung nur um einen konstanten Term, der + bei der Variation verschwindet. Deshalb bleiben die Bewegungsgleichungen unverändert. + \end{enumerate} +\end{aufgabe} + +\end{document}