Math 6410 • ODE

The simple pendulum consists of a point particle of mass \(m\) suspended from a fixed
point by a massless rod of length \(L\), which is allowed to swing in a vertical plane.
If friction is ignored then the equation of motion is
\[
  \ddot{x}+\omega^2\sin x=0,\qquad \omega^2=\frac{g}{L},
\]
where \(x\) is the angle of inclination of the rod with respect to the downward vertical
and \(g\) is the gravitational constant.
\begin{enumerate}
\item[(a)] Using conservation of energy, show that the angular velocity of the pendulum
satisfies
\[
  \dot{x}=\pm\sqrt{2}(C+\omega^2\cos x)^{1/2},
\]
where \(C\) is an arbitrary constant. Express \(C\) in terms of the total energy of the
system.
\item[(b)] Sketch the phase diagram of the pendulum equation in the \((x,\dot{x})\)-plane.
Illustrate the one-parameter family of curves given by part (a) for different values
of \(C\). Take \(-3\pi\le x\le3\pi\). Indicate the fixed points of the system and the
separatrices - curves linking the fixed points. Give a physical interpretation of the
underlying trajectories in the two distinct dynamical regimes \(\abs{C}<\omega^2\) and
\(\abs{C}>\omega^2\).
\item[(c)] Show that in the regime \(\abs{C}<\omega^2\), the period of oscillations is
\[
  T=4\sqrt{\frac{L}{g}}K(\sin x_0/2),
\]
where \(\dot{x}=0\) when \(x=x_0\) and \(K\) is the complete elliptic integral of the first
kind, which is defined by
\[
  K(\alpha)=\int_0^{\pi/2}\frac{1}{\sqrt{1-\alpha^2\sin^2u}}\dd u.
\]
\item[(d)] For small amplitude oscillations, the pendulum equation can be approximated
by the linear equation
\[
  \ddot{x}+\omega^2x=0.
\]
Solve this equation for the initial conditions \(x(0)=A\), \(\dot{x}(0)=0\) and sketch
the phase-plane for different values of \(A\). Compare with the phase-plane for the
full nonlinear equation in part (b).
\end{enumerate}