Through Isaac Newton's Discoveries, the swinging of a single pendulum is a model that is possible to predict. However, by the understandings of Chaos Theory, a double-pendulum is not as easy of a model to predict. The double-pendulum bounces in an unpredictable pattern. The initial conditions allow some prediction in the near future, but the long-term prediction is near impossible as initial conditions are changed ever-so-slightly.
Thus, the Double Pendulum is a useful tool to show to Chaos Theory as it shows that tiny changes in parameters (i.e., gravity, mass of both objects, lengths of each string) effect the positions and angles of each mass and their respective line. The calculations come from the equations and uses of this webpage which dives deeper into the double pendulum. Although a single pendulum is predictable, the double pendulum is very difficult, as it follows a non-linear equation.
Some questions for exploring:
The orange mass is dependent upon the green mass. The orange mass follows the lines of the circles (as shown when you select "Show Circle Limits"). The green mass does not go beyon the green circle limits, but the green mass can go inside the orange circle. The trace of the green mass shows a non-linear pattern. It's pattern is highly dependent upon each variable. As such, any change causes the trace of the green mass to change significantly, especially as time increases.
Isaac Newton asserted that the concepts of law of motion and initial conditions are separate. Newton simplified the model to calculate the planets' trajectories; he developed differential calculus and gravitational law. However, Laplace was able to further explain Newton's findings through differential equations.1
In the nineteenth century, scientists were not able to fully determine the stability of the solar system.2 However, with the use of computers and machine learning, the prediction of the orbit of planets has become more accurate. Computers have allowed man to send rovers to Mars.
According to Chaos Theory, at any moment, the prediction of planetary movements could change with a tiny, unexpected factor, leading to a much different output. Thus, predicting the orbit 500 million years in advanced is nearly impossible.
