# Equations of motion of coupled pendulum using the lagrangian method

Question
Obtain the equations of motion of using the lagrangian method.
Solution
Consider a system of coupled pendulums as shown below in the figure

The displacement of A is and B is , condition being < .

In such state the spring gets stretched. The lengths of the strings of both the pendulums are same (say l).

The angular displacement of A is and that of B is .
Therefore

As the spring gets stretched, it is clear from the figure that restoring force works along the direction of displacement and opposite to the direction of displacement .

Now A and B at zero potential level, the total potential energy of the system is given as

Where m is the mass of each one of the bob and k is the spring constant.

Since and are small so,

Neglecting the higher powers other than squares of and the expression of potential energy can be written as

Also the kinetic energy of whole system is

Hence Lagrangian L would be

Now

Hence Lagrangian equation in terms of is

Also,

Hence Lagrangian equation in terms of is

The equation of motion for given system are