Oscillatory–ballistic motion regularities of a gravitational pendulum

We simulate the motion of a gravitational pendulum that has initial angular amplitude larger than 90∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document} and smaller than 180∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}, and loses energy at each change from ballistic to oscillatory motion when the string is suddenly tensed (we name this event collision). Simulation is based on a velocity Verlet algorithm that is implemented in a numerical code. The numerical simulation of motion as function of time is checked against an analytical code that describes the trajectory. The string tension expression that respects the velocity Verlet algorithm requirements is identified and a criterion for collision occurrence is introduced. An interesting band-like structure of the number of collisions as function of the initial amplitude and damping modelling is obtained.