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 \[ \ddot {x}+\omega ^2\sin x=0,\quad \omega ^2=\frac {g}{L}, \] where \(x\) is the angle of inclination of the rod with respect to the downward vertical and \(g\) is the gravitational constant.
- 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.
- 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 \(-3\pi \le x \le 3\pi \). 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 \(|C|<\omega ^2\) and \(|C|>\omega ^2\).
- Show that in the regime \(|C|<\omega ^2\), 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 \[ K(\alpha )=\int _0^{\pi /2}\frac {1}{\sqrt {1-\alpha ^2\sin ^2 u}}\,du. \] HINT: Derive an integral expression for \(T\) and then perform a change of variables \[ \sin u=\frac {\sin (x/2)}{\sin (x_0/2)}. \]
- 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).