Is the Motion Simple Harmonic? - 2

The Lagrangian for this system is:

$L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} k \Big( \sqrt{a^2 + x^2} - a \Big)^2$

The equation of motion is:

$\frac{d}{dt } \frac{\partial{L}}{\partial{\dot{x}}} = \frac{\partial{L}}{\partial{x}}$

Evaluating results in:

$m \ddot{x} = - k x \Big(1 - \frac{a}{\sqrt{a^2 + x^2}} \Big)$

Taylor series approximation for small $x$ :

$\sqrt{a^2 + x^2} \approx a + \frac{x^2 }{2 a}$

Plugging into the previous expression and simplifying results in:

$m \ddot{x} \approx - k x \Big(\frac{x^2}{2 a^2 + x^2} \Big) \approx -\frac{k x^3}{2 a^2}$

Using small- $x$ approximations, the double-dot term is proportional to the cube of the displacement. Thus, the expression does not represent SHM.