Double pendulum

Question 

In case of a double pendulum find the expression for the kinetic energy of the system

Solution

We take a simple case where lengths and masses are same.see below in the figure

Here on being displaced the co-ordinates of two pendulums are

$\begin{array}{l}{x_1} = l\sin {\theta _1}\\{y_1} = l\cos {\theta _1}\end{array}$

For the first pendulum where ${\theta _1}$ is the angle through which the first pendulum have been displaced.

For second pendulum

$\begin{array}{l}{x_2} = l\sin {\theta _1} + l\sin {\theta _2}\\{y_2} = l\cos {\theta _1} + l\cos {\theta _2}\end{array}$

Where ${\theta _2}$ is the angle through which second pendulum has been displaced.

The total kinetic energy of the system is given by the expression

$T = \frac{1}{2}m(\dot x_1^2 + \dot y_1^2) + \frac{1}{2}m(\dot x_2^2 + \dot y_2^2)$

Now

And

Which is the required result