I'm finding the equations of motion of a mass attached to four springs in a box. See picture:
In the prompt, we're instructed to use "the small-oscillations approximation, and neglect terms of order $\frac{x^2}{a^2}$ , $\frac{y^2}{a^2}$ , and $\frac{xy}{a^2}$". This all makes perfect sense to me.
Using both force diagrams and the Lagrangian approach, I find the equations of motion. I have the solution, but I do not see how it is possible to reach that solution.
For example, let's find the x-component of the force from the spring at the "top" of the box. The length of the spring for an arbitrary x, y is $\sqrt{x^2 + (a-y)^2}$, and so our total force vector is $F_1$ = $K_2\left(a-\sqrt{x^2 + (a-y)^2}\right)$. And taking the x-component we have:
$$ F_{1x} = K_2\left(a-\sqrt{x^2 + (a-y)^2}\right) \frac{x}{\sqrt{x^2 + (a-y)^2}} $$
And I am told from the solutions that $F_{1x} \approx 0$. I cannot see how this is possible. I've tried using the approximation $(1+x^2)^{-1/2} \approx (1-\frac{1}{2}x^2)$, but it seems no matter what I do I fail to reach 0.
Does anyone see how small angle approximation can lead to getting $F_{1x} = 0$ here?