Cococubed.com The pendulum
Home Astronomy Research    Radiative Opacity    2024 Neutrino Emission from Stars    2023 White Dwarfs & 12C(α,γ)16O    2023 MESA VI    2022 Earendel, A Highly Magnified Star    2022 Black Hole Mass Spectrum    2021 Skye Equation of State    2021 White Dwarf Pulsations & 22Ne    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      Adiabatic white dwarfs      Cold neutron stars      Stellar opacities      Neutrino energy loss rates      Ephemeris routines      Fermi-Dirac functions      Polyhedra volume      Plane - cube intersection      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      Hydrodynamic test cases      Galactic chemical evolution      Universal two-body problem      Circular and elliptical 3 body      The pendulum      Phyllotaxis      MESA      MESA-Web      FLASH      Zingale's software      Brown's dStar      GR1D code      Iliadis' STARLIB database      Herwig's NuGRID      Meyer's NetNuc AAS Journals    2024 AAS YouTube    2024 AAS Peer Review Workshops 2024 ASU Energy in Everyday Life 2024 MESA Classroom Outreach and Education Materials Other Stuff:    Bicycle Adventures    Illustrations    Presentations 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 0.55in {\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 $\kappa$ in the above expressions, and applying properties of the Jabobi elliptic functions: \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} The tool pendulum.tbz implements this complete analytical solution to the classic nonlinear pendulum. The solution is valid for any initial conditions and holds if the pendulum swings over or not. Phase diagram of the pendulum. Family of $\theta$ (red hues) and $\dot{\theta}$ (blue hues) solutions for $E_0 \lt E_p$ (pendulum does not swing over).
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