Derivation of the Euler Lagrange Equation

Question

I'm self studying a little bit of physics at the moment and for that I needed the derivation of the Euler Lagrange Equation. I understand everything but for a little step in the proof, maybe someone can help me. That's were I am: $$ \frac{dJ(\varepsilon=0 )}{d\varepsilon } = \int_{a}^{b}\eta(x)\frac{\partial F}{\partial y}+\eta'(x)\frac{\partial F}{\partial y'}dx = 0 $$ Then the second term is integrated by parts: $$ \frac{dJ(\varepsilon =0)}{d\varepsilon } = \int_{a}^{b}\eta(x)\frac{\partial F}{\partial y}dx + \left [ \frac{\partial F}{\partial y'}\eta(x) \right ]_a^b-\int_{a}^{b}\frac{d}{dx}(\frac{\partial F}{\partial y'})\eta(x) = 0 $$ And the equation is simplified to: $$ \frac{dJ(\varepsilon =0)}{d\varepsilon } = \int_{a}^{b}\frac{\partial F}{\partial y}-\frac{d}{dx}(\frac{\partial F}{\partial y'})\eta(x)dx = 0 $$ What I don't understand is why you just can omit the $$ \left [ \frac{\partial F}{\partial y'}\eta(x) \right ]_a^b $$ Why does that equal zero, but the integral following it which also goes from a to b isn't left out? I think it's pretty obvious, but I'm just to stupid to see it. I'd appreciate if someone could help me!

Answer 1

In the Euler-Lagrange equation, the function $\eta$ has by hypothesis the following properties:

• $\eta$ is continuously differentiable (for the derivation to be rigorous)
• $\eta$ satisfies the boundary conditions $\eta(a) = \eta(b) = 0$.

In addition, $F$ should have continuous partial derivatives.

This is why $\left [ \frac{\partial F}{\partial y'}\color{red}{\eta(x)} \right ]_a^b$ simplifies to $0$.