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\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Homework \#6, Rev. B, due Tuesday, 2018-10-16 }
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% {\bf 1.} 

% {\bf \quad d)} Symplectic integrator: conserve the Hamiltonian, which in this case has the interpretation of total energy of the system.


{\bf 1. Pendulum}  A mass $m$ swings at the end of a massless rod $\ell$ rotating about a fixed point at the top.

% Reformulate the pendulum in terms of Lagrangian and then Hamiltonian. Simulate in Matlab.  Animate a plot of the phase space while you integrate the equations of motion in two panels on the same figure. [bonus:] What are the canonical action-angle coordinates in theis case?  Hint: the canonical angle is neither the angle in phase space nor the angle of the pendulum.

{\bf \quad a)} Calculate the Lagrangian $\mathcal L(\dot\theta,\theta,t)$ and resulting equation of motion for the undamped case.  Express it in terms of the natural frequency $\omega_{0}=\sqrt{g/\ell}$ at small angles.  [bonus:] Calculate the exact trajectory, starting from rest at angle $\theta_{0}$, in terms of the Jacobi elliptic function $sn(u;m)$ and calculate the frequency $\omega(\theta_{0})$.

{\bf \quad b)} Calculate the canonical momentum $p_{\theta}$ and Hamilton's equations for the undamped case.  Calculate the maximum velocity of $v_{1}$ of the pendulum, starting from rest at the top, using $m=1~\q{kg}$ and $\ell=1~\q{m}$.  Plot trajectories in phase space starting from equilibrium $\theta=0$ with initial velocity $v_{k}=k v_{1}$ for $k=0, \pm\frac15, \pm\frac25,\frac 35, \ldots 2$.  You can obtain a more complete flow of phase space by starting from more equilibrium points: $\theta_{0}=0,\pm 2\pi, \pm 4\pi$.

{\bf \quad c)} Recalculate the trajectories, adding a frictional force $F=-bv$, starting with $b=1~\q{kg/s}$, normalized to $2\beta=b/m$.  Determine the critical damping coefficient $b_{c}$, above which oscillations entirely disappear.  Repeat the phase space plots for $b=b_{c}/2, b_{c}$, and $2b_{c}$.

{\bf \quad d)} Recalculate the trajectories, adding a driving force $F=-F_{0}\cos(\omega t)$ with $F_{0}=\gamma mg$, and $\gamma=1$ for this problem.  Repeat the trajectories $\theta(t)$ and phase space diagrams $[p_{\theta},\theta](t)$ for the same values of $b$ as part c).

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