A blog of Python-related topics and code.
This post is based on Problem 2.22 from Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995).
The Belousov-Zhabotinsky (BZ) reaction is a classical example of a non-equibrium chemical oscillator in which the components exhibit periodic changes in concentration.
For the purposes of this article, the harmonically-driven pendulum is one whose anchor point moves in time according to $x_0(t) = A\cos\omega t$. As with previous posts, the position of the pendulum bob with time can be described using Lagrangian mechanics. In a coordinate system with the pendulum anchor initially at $(0,0)$ and the $y$-axis pointing up, the components of the bob position and velocity as a function of time are:
Following on from the previous post on the double pendulum, here is a similar Python script for plotting the behaviour of the "spring pendulum": a bob of mass $m$ suspended from a fixed anchor by a massless spring.
In classical mechanics, a double pendulum is a pendulum attached to the end of another pendulum. Its equations of motion are often written using the Lagrangian formulation of mechanics and solved numerically, which is the approach taken here. The dynamics of the double pendulum are chaotic and complex, as illustrated below.