Analytical study of the critical behavior of the nonlinear pendulum

The dynamics of a simple pendulum consisting of a small bob and a massless rigid rod has three possible regimes depending on its total energy $E$⁠: Oscillatory (when $E$ is not enough for the pendulum to reach the top position), “perpetual ascent” when $E$ is exactly the energy needed to reach the top, and nonoscillatory for greater energies. In the latter regime, the pendulum rotates periodically without velocity inversions. In contrast to the oscillatory regime, for which an exact analytic solution is known, the other two regimes are usually studied by solving the equation of motion numerically. By applying conservation of energy, I derive exact analytical solutions to both the perpetual ascent and nonoscillatory regimes and an exact expression for the pendulum period in the nonoscillatory regime. Based on Cromer’s approximation for the large-angle pendulum period, I find a simple approximate expression for the decrease of the period with the initial velocity in the nonoscillatory regime, valid near the critical velocity. This expression is used to study the critical slowing down, which is observed near the transition between the oscillatory and nonoscillatory regimes.