Introduction

The double pendulum is a physical system that demonstrates chaotic behavior. Unlike a single pendulum, which consists of a single mass swinging from a fixed point, the double pendulum comprises two connected pendulums, each with its own mass and length, swinging from a common pivot point. Despite its simple structure, the double pendulum's motion can be incredibly complex and unpredictable, making it a subject of interest in physics and mathematics.

Numerical Analysis: Runga-Kutta methods

In the initial stages of the simulation, the approach was to calculate the angular acceleration at each point and directly apply it to update the velocity. While this method produced reasonable results, a noticeable discrepancy in the total energy of the system became apparent over time.

The discrepancy in the total energy arose due to the nature of the double pendulum's motion. The system's dynamics were governed by non-linear equations, which meant that small changes in the initial conditions or calculation inaccuracies can lead to significant deviations in the predicted motion.

To address this challenge, numerical analysis techniques were employed to approximate solutions to the non-linear equations. These techniques involved breaking down the motion of the double pendulum into small time steps and iteratively calculating the positions and velocities of the pendulum masses at each step. By using these approximations, the motion of the double pendulum could be simulated with greater accuracy than would have been possible by simply reducing the step size in the calculation. Additionally, numerical analysis allows for more efficient computation of the system's behavior, making it a valuable tool in studying complex dynamic systems like the double pendulum.

For this simulation, the chosen numerical analysis technique was the Runge-Kutta (RK4) method. Unlike the Euler method, which only considers the first derivative in its calculations, RK4 takes into account the first four derivatives. This means that RK4 provides a more accurate approximation of the system's behavior by accounting for higher-order derivatives.

Read more about the Runga-kutta methods here.