The pendulum


Astronomy research
Software instruments
   Stellar equation of states
   EOS with ionization
   EOS for supernovae
   Chemical potentials
   Stellar atmospheres

   Voigt Function
   Jeans escape
   Polytropic stars
   Cold white dwarfs
   Hotter white dwarfs

   Cold neutron stars
   Stellar opacities
   Neutrino energy loss rates
   Ephemeris routines
   Fermi-Dirac functions

   Galactic chemical evolution
   Coating an ellipsoid

   Nuclear reaction networks
   Nuclear statistical equilibrium
   Laminar deflagrations
   CJ detonations
   ZND detonations

   Fitting to conic sections
   Unusual linear algebra
   Derivatives on uneven grids
   Pentadiagonal solver
   Quadratics, Cubics, Quartics

   Supernova light curves
   Exact Riemann solutions
   1D PPM Hydrodynamics
   Verification problems
   Plane - Cube Intersection

   The pendulum


   Zingale's software
   Brown's dStar
   GR1D code
   Iliadis' STARLIB database
   Herwig's NuGRID
   Meyer's NetNuc
Bicycle adventures

AAS Journals
2017 MESA Marketplace
2017 MESA Summer School
2017 ASU+EdX AST111x
Teaching materials
Education and Public Outreach

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:

Phase diagram of the pendulum.

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

Please cite the relevant references if you publish a piece of work that use these codes, pieces of these codes, or modified versions of them. Offer co-authorship as appropriate.