Author Topic: HA10-P2  (Read 469 times)

Victor Ivrii

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Chi Ma

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Re: HA10-P2
« Reply #1 on: December 08, 2015, 01:44:06 AM »
Let $L = \frac{1}{\sqrt{2gu}} \sqrt{1+u^{\prime\,2}}$. Hamiltonian $H=u' L_{u'}-L=constant$ implies

\begin{equation} u' =  \sqrt{\frac{2A-u}{u}} \label{u} \end{equation} for some constant $A$. Reparameterize  $u = A - A\cos\theta$ to obtain the solution to ($\ref{u}$).
\begin{equation} x =  A (\theta - \sin \theta) + C \end{equation}

Condition $u(0)=0$ implies that $C=0$. The solution is a parametric cycloid:
\begin{equation} x =  A (\theta - \sin \theta) \end{equation}
\begin{equation} u =  A (1 - \cos \theta) \end{equation}
« Last Edit: December 08, 2015, 01:00:26 PM by Chi Ma »

Victor Ivrii

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Re: HA10-P2
« Reply #2 on: December 22, 2015, 02:52:34 AM »
OK. But need more details in the end