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{\bf SOLUTIONS MIDTERM PHY 4936 (Fall 2011)}\\ 
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\centerline{\bf PROBLEM 2}\bigskip

\noindent 2.1 In sperical coordinates with $r=R=const$
$$ L\ =\ \frac{m}{2}\,R^2\,\left(\dot\phi^2\sin^2\theta
      + \dot\theta^2\right) - m\,g\,R\,\cos\theta\ .$$
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\noindent 2.2 The conservations laws are
$$ E\ =\ \frac{m}{2}\,R^2\,\left(\dot\phi^2\sin^2\theta
      + \dot\theta^2\right) + m\,g\,R\,\cos\theta $$
energy conservation and
$$ p_{\phi}\ =\ \frac{\partial L}{\partial \dot\phi}\ 
             =\ m\,R^2\,\dot\phi\,\sin^2\theta\ =\ M_z\ .$$
angular momentum conservation.
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\noindent 3.3 Special solution $\theta(t)=\theta_0=const$:
$$ \dot\phi\ = \frac{M_z}{m\,R^2\,\sin^2\theta_0}\ =\ \omega_0
   \ \Rightarrow\ \phi(t)\ =\ \omega_0\,t + \phi_0\ . $$
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