I'm teachin myself the basics of Calculus of variations. So far I know how to calculate the Euler Lagrange equation for simple functionals.
I'm now stuck on how to compute the total differentiation of the following problem:
$$I[y]=\int_0^1 (y\frac{dy}{dx})^2 -\lambda y^2 \ dx$$
To calculate the Euler Lagrange equation I have the following:
$$\frac{d}{dx}(2y^2y')-2y^2\frac{dy}{dx}+2\lambda y=0$$.
Is this correct? If so I'm unsure of how to evaluate the total differentiation part. That is taking the total derivative in this case.
Best Answer
For the $I$ you gave above, $f(x,y,y')=(yy')^2-\lambda y^2$, hence the E-L equation for $I$ is $$ 2y\left(\frac{dy}{dx}\right)^2-2\lambda y-\frac{d}{dx}\left(2y^2\frac{dy}{dx}\right)=0 $$ Your only mistake is that $\frac{\partial f}{\partial y}=2\color{red}{y(y')^2}-2\lambda y$, not $2\color{red}{y^2y'}-2\lambda y$ as you have written. Does this help you go from here?
Edit. Since $f$ does not depend upon $x$, we can simplify the problem.
Proof. Just calculate the derivative $$ \frac{d}{dx}\left(y'\frac{\partial f}{\partial y'}-f\right)=y''\frac{\partial f}{\partial y'}+y'\frac{d}{x}\frac{\partial f}{\partial y'}-\frac{\partial f}{\partial x}-y'\frac{\partial f}{\partial y}-y''\frac{\partial y}{\partial y'} $$ Since $f$ is independent of $x$, $\frac{\partial f}{\partial x}=0$, and we can take out a factor of $y'$ $$ \frac{d}{dx}\left(y'\frac{\partial f}{\partial y'}-f\right)=y'\left(\frac{d}{dx}\frac{\partial f}{\partial y'}-\frac{\partial f}{\partial y}\right) $$ which is clearly equal to $0$ by E-L. QED
This means that $$ y'\frac{\partial f}{\partial y'}-f=\text{constant} $$ Applying this to your function $I$ gives $$ \begin{align*} y'(2y^2y')-(yy')^2+\lambda y^2&=C \\ y^2\left(\left(\frac{dy}{dx}\right)^2+\lambda\right)&=C \\ \frac{dy}{dx}&=\sqrt{\frac{C}{y^2}-\lambda} \end{align*} $$