### Oscillations

Potential energy $U=mgy=mgax^2$. Kinetic energy
$K=(m/2) (\dot{x}^2+\dot{y}^2)=\frac{m}{2}(1+4a^2x^2)\dot{x}^2$
When $ax\ll 1$ we have $K\approx \frac{m}{2} \dot{x}^2$ and $\omega^2=2ga$.