Synopsis: The pendulum project was one that I undertook in university to create an inertial double pendulum simulator. The goal was to understand chaos, and to understand how chaos adapts and evolves in a controlled environment. On this page, we talk about point mass pendulum (where the center of gravity is located at the very end of the arm. The report associated with this assignment was based on an inertial double pendulum, where the center of mass is distributed along the entire arm. I have changed it up here for two reasons. Firstly, to make it unique to the assignment, so if some future student goes to copy my work, they will at least have to change some of the maths to get it to work, and secondly, it is far easier to do the maths on a point mass pendulum then it is to to do on an inertial pendulum.<\/p>\n\n\n\n
In the world of physics, knowing exact initial conditions will always result in knowing exactly how an object will behave. In most cases, knowing approximate initial conditions can lead to an approximate solution. In other cases, such as that of the double pendulum, a tiny shift of any initial conditions will cause the trajectory of the pendulum to change drastically over time. The moment there is any difference in conditions, the result will be completely unique. <\/p>\n\n\n\n
Why undertake the study of something like this? Well, two reasons, it is an excellent example of lagrangian mechanics, and chaotic objects inspire fascination in humans. We feel like we can predict it, understand it, and then the universe taunts us, spitting in our faces and doing exactly what we don’t expect and being chaotic. As Einstein said, “God does not play dice with the universe,” and chaotic systems feel like dice. Well, I see this as me fighting back against the universe, trapping its chaos in a simulation and forcing it to bow to my whim, my control, all the while pulling data from it, and performing analysis on it. The study of chaos is a traditional man vs god story, within a wrapper of fascinating mathematics and science. <\/p>\n\n\n\n
Before doing the maths on this, we need to understand what is going on, and define our variables. We are going to use the following image to define our variables for our maths. We can see from the diagram below the angles and values that we will be using. <\/p>\n\n\n\n