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{\bf SOLUTIONS MIDTERM PHY 4936 (Fall 2011)}\\ 
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\centerline{\bf PROBLEM 1}\bigskip

\noindent 1.1 The Lagrangian is
$$ L=\frac{1}{2}m\dot{x}^2-\frac{1}{2}k\,x^2 \;.  $$
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\noindent 1.2 Euler-Lagrange equation
\begin{eqnarray} \nonumber
  \frac{d~}{dt}\,\frac{\partial L}{\partial\dot x}\
  &=& \frac{\partial L}{\partial x} \\ \nonumber
  m\,\ddot x &=& - k\,x \\ \nonumber
  \ddot x &=& - \frac{k}{m}\,x\ =\ - \omega^2\,x
\end{eqnarray}
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\noindent 1.3 Solution:
$$ x(t)\ =\ x_0\,\cos(\omega\,t) + 
          \frac{\dot x_0}{\omega}\,\sin(\omega\,t)\ .$$
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