# Proving that the Euler-Lagrange Equation has no solution [closed]

I'm trying to show that the Euler - Lagrange equation for the functional $$I(y)=\int_{a}^{b} y\:dx$$

subject to $$y(0)=y(1)=0$$ has no solutions.

The Euler - Lagrange equation states that: $$\frac{d}{dx}\frac{\partial F}{\partial y'}-\frac{\partial F}{\partial y}=0$$

For this specific problem, $$F=y$$. Hence, $$\frac{\partial F}{\partial y'}=0$$. So: $$-\frac{\partial F}{\partial y}=0$$

I don't understand why there's no solution when one could say that $$y=c$$, where $$c$$ is a constant. I understand that $$I(y)$$ does not have an extremum but how do I prove that the Euler Lagrange equation has no solution?

## closed as off-topic by JMac, GiorgioP, Kyle Kanos, ZeroTheHero, user191954 Mar 9 at 14:33

• This question does not appear to be about physics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

• What is $\frac{\partial F}{\partial y}$ equal to? – jacob1729 Mar 6 at 15:24
• @jacob1729 I mean, $F=y$, so the derivative would just be $1$ and that would lead to $-1=0$ ( as posted below). Does that prove that the Euler - Lagrange equation has no solution ? – Jim Β Mar 6 at 15:34
• @JimΒ I made my comment before that post. Do you see why that is a contradiction then? – jacob1729 Mar 6 at 16:13
• @jacob1729 yeah I do . Thank you and the OP of the answer below for your input. – Jim Β Mar 6 at 16:14
• Related Math.SE question: math.stackexchange.com/q/3136864/11127 – Qmechanic Mar 7 at 2:10

## 1 Answer

$$\partial F/\partial y$$ is not an unknown, it is equal to 1, so your equation is $$-1=0$$.