Derive the equations of motion for a simple harmonic oscillator


 * 1) Find the Lagrangian that describes a simple harmonic oscillator: \(\mathcal{L}(x(t), \dot{x}(t))=\frac{1}{2}m\dot{x}^2 - \frac{1}{2}m\omega^2 x^2\)
 * 2) Input this into the Euler-Lagrange equation, \(\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{x}}-\frac{\partial\mathcal{L}}{\partial x} = 0\): \(\frac{d}{dt}\frac{\partial}{\partial\dot{x}} \left (\frac{1}{2}m\dot{x}^2 - \frac{1}{2}m\omega^2 x^2\ \right ) -\frac{\partial}{\partial x} \left ( \frac{1}{2}m\dot{x}^2 - \frac{1}{2}m\omega^2 x^2 \right ) = 0\)
 * 3) Solve the derivative: \(m\ddot{x} + m\omega^2 x = 0\)
 * 4) Simplify: \(\ddot{x} + \omega^2 x = 0\)