Classical Dynamics Any help would be greatly appreciated

[misslogica]

Homework Statement

A mass m₁ is attached to a fixed spring on a horizontal surface and attached across a pulley (ignore the pulley mass) to another freely hanging m₂. Write the Lagrangian in terms of a single parameter. Find the equation of motion and determine the frequency of oscillation.

Homework Equations

L = T-U

The Attempt at a Solution

 

[hilbert2]

The potential energy of the system comes from the stretching of the spring and from the gravitational potential energy of the hanging object. You should be able to write these contributions in terms of one coordinate $x$ which gives the spring's extention relative to equilibrium. The velocities of the objects are equal, as the objects are connected to each other. Therefore the kinetic energy is just $\frac{1}{2}(m_{1}+m_{2})v^{2}$, where $v=\dot{x_ {1}}=\dot{x_{2}}$ is the velocity of the objects.