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The pendulum

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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.

Referring to the diagram in the upper left, the pendulum obeys the conservation of angular momentum \begin{equation} \ddot{\theta} + g/l \ \sin(\theta) = 0 \hskip 0.5in \theta(t_0) = \theta_0 \hskip 0.5in \dot{\theta}(t_0) = \dot{\theta}_0 \ . \label{eq1} \tag{1} \end{equation} The analytical solution when the pendulum has enough energy to swing over is \begin{equation} \begin{split} A & = {\rm sgn}(\dot{\theta}) k \omega [t - t_0] + F(\sin^{-1}(k_0),\kappa) \\ \theta & = 2 \sin^{-1}({\rm sn}(A,\kappa)) \cdot {\rm sgn}({\rm cn}(A,\kappa)) \\ \dot{\theta} & = {\rm sgn}(\dot{\theta}) \ \sqrt{E_0} \ {\rm dn}(A,\kappa) \end{split} \label{eq2} \tag{2} \end{equation} where \begin{equation} \begin{split} {\rm sgn}(\zeta) & = 1 \ {\rm for} \ (\zeta) \ge 0 \ ; -1 \ {\rm for} \ (\zeta) \lt 0 \hskip 0.25in {\rm ! \ signum \ function } \\ \omega & = \sqrt{g / l} \hskip 2.3in {\rm ! \ angular \ frequency} \\ k_0 & = \sin(\theta_0 / 2) \hskip 1.95in {\rm ! \ sine \ half \ angle} \\ E_p & = 4 \omega^2 \hskip 2.45in {\rm ! \ maximum \ potential \ energy} \\ E_0 & = \dot{\theta}_0^2 + E_p \sin^2(\theta_0) \hskip 1.35in {\rm ! \ total \ energy } \\ k & = \sqrt{E_0 / E_p} \ge 1\\ \kappa & = 1/k \end{split} \label{eq3} \tag{3} \end{equation} $g$ is the gravitational acceleration, $F(\phi,m)$ is the Legendre form of the first incomplete elliptic integral, and ${\rm sn}(u,m)$, ${\rm cn}(u,m)$, and ${\rm dn}(u,m)$ are the Jacobi elliptic functions.

The solution when the pendulum does not have enough energy to swing over (now $k \le 1$ and $\kappa \gt 1$) is found by swapping $k$ and $\kappa$, applying properties of the Jabobi elliptic functions, and simplifying: \begin{equation} \begin{split} A & = {\rm sgn}(\dot{\theta}) \omega [t - t_0] + F(\sin^{-1}(\kappa k_0),k) \\ \theta & = 2 \sin^{-1}(k \ {\rm sn}(A,\kappa)) \\ \dot{\theta} & = {\rm sgn}(\dot{\theta}) \ \sqrt{E_0} \ {\rm cn}(A,\kappa) \end{split} \label{eq4} \tag{4} \end{equation} This code implements this complete analytical solution to the classic nonlinear pendulum. The solution valid for any initial conditions and holds if the pendulum swings over or not. This code generated these plots:


image
Phase diagram of the pendulum.



image
Family of $\theta$ (red hue) and $\dot{\theta}$ (blue hue) solutions for $E_0 \lt E_p$ (pendulum does not swing over).
 



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