# Swinging Sticks Mathematics

## Double Pendulums

- x = Horizontal Coordinate
- y = Vertical Coordinate
- θ = Angle of Pendulum
- L = Length
- m = mass
- g = gravity (9.81 m/s
^{2})

A double pendulum is a pendulum that has another one attached to the end of it.

For the most part, the motion is seemingly chaotic however there is actually mathematic beauty behind the Swinging Sticks

The Swinging Sticks is considered a simple or compound double pendulum because it operates solely in a 2D plain.

## Swinging Sticks Physics

The Swinging Sticks equations for double pendulums are solved using the Runge-Kutta methods of numerical analysis. This technique was developed in the early 1900's by mathematicians C. Runge and M.W. Kutta

Calculate all the following equations for

x_{1}, x_{2}, y_{1}, y_{2}

1) Calculate the position of masses

x = L sin θ

2) Next take the derivative to find the velocity

x_{1}^{'} = θ_{1}^{'} L_{1} cos θ_{1}

x_{2}^{'} = x_{1}^{'} + θ_{2}^{'} L_{2} cos θ_{2}

3) Take another derivative to find the acceleration

x_{1}^{''} = −θ_{1}^{'2} L_{1} sin θ_{1} + θ_{1}^{''} L_{1} cos θ_{1}

x_{2}^{''} = x_{1}^{''} − θ_{2}^{'2} L_{2} sin θ_{2} + θ_{2}^{''} L_{2} cos θ_{2}

0 = vertical downwards

Counter-clockwise = positive

Click here to learn more about the physics behind the Swinging Sticks Kinetic Energy Sculpture.