I'm studying the second variation in Calculus of Variation.
For functional $\int_{a}^{b}f(t,y,y')$ ($y(a), y(b)$ are fixed constants), the second variation $\delta^2J[\eta,y]$ is given by $$\int_{a}^{b}\frac{1}{2}f_{y'y'}\eta'^2 + \frac{1}{2}(f_{yy}-\frac{d}{dx}f_{yy'})\eta^2 dx$$ where $y$ is chosen to be an extremal curve, i.e. the one that satisfies the Euler-Lagrange equation.
And the Jacobi accessory equation is given by $$\frac{d}{dx}(f_{y'y'}u') - (f_{yy}-\frac{d}{dx}f_{yy'})u=0$$ which coincides (in form) with the Euler lagrange equation for the functional $G[u]:=\delta^2J[u,y]$. Namely, if we consider the fucntional $\delta^2[\eta,y]$ as a functional of $\eta$, writing down the EL eqn. will give you the Jacobi accessory equation.
So, why is that? Why do they coincide?