Find the Gateaux differential for the functional $ S[y]=y(0)+\frac{1}{2}\int_{0}^{1}(4y'^2+x^2y^2)dx, y(1)=2 $ and use it to derive the Euler-Lagrange equation. Be sure to specify all boundary conditions that the stationary path must satisfy.
Here's my work:
We have $ S[y+\epsilon h]=y(0)+\epsilon h(0)+\frac{1}{2}\int_{0}^{1}[4(y'+\epsilon h')^2+x^2(y+\epsilon h)^2]dx $.
Hence $ \frac{d}{d\epsilon}S[y+\epsilon h]=h(0)+\frac{1}{2}\int_{0}^{1}[8(y'+\epsilon h')h'+2x^2(y+\epsilon h)h]dx $.
Thus $ \bigtriangleup S[y, h]=h(0)+\frac{1}{2}\int_{0}^{1}(8y'h'+2x^2yh)dx $,
so the Gateaux differential is $ \bigtriangleup S[y, h]=h(0)+\int_{0}^{1}(4y'h'+x^2yh)dx $.
And now I'm stuck. How should I use this Gateaux differential to derive the associated Euler-Lagrange equation? And how to specify all boundary conditions that the stationary path must satisfy?
Best Answer
First note that for $y + \varepsilon h$ to be admissable, $h(1)$ must be $0$. Now take the last expression and integrate by parts $$\delta S[y;h] = h(0) + \int_0^14y'h' + x^2yh~\mathrm{d}x = h(0) + \int_0^1(-4y'' + x^2y)h~\mathrm{d}x + 4y'(1)h(1) - 4y'(0)h(0).$$ For $y$ to be an extremal of this functional, this quantity must be zero for every admissible $h$. By choosing $h$ with compact support on $(0,1)$, we can see that the integral must be zero, which implies that $-4y'' + x^2y = 0$ identically on $(0,1)$. However, some admissible $h$ may be nonzero at $0$, so more conditions on $y$ are required to ensure the boundary terms are zero.
First notice that $h(1)$ is zero via the definition of admissible perturbations. The remaining boundary terms are $$h(0) - 4hy'h(0) = (1 - 4y'(0))h(0).$$ Since $h(0)$ is arbitrary, this can only be zero for all $h$ if $1-4y'(0) = 0\implies y'(0) = 1/4.$ The complete Euler-Lagrange equation is then $$-4y'' + x^2y=0, \\ y'(0) = \frac{1}{4}, \quad y(1) = 2.$$