# Calculus of variations

1. Oct 28, 2014

### jimmycricket

1. The problem statement, all variables and given/known data
Find the extremal for the case
$$\int_a^b y^2(1+(y')^2) \, dx$$
where $y(a)=y_{0}, y(b)=y_{1}$

2. Relevant equations

3. The attempt at a solution
Using the Euler-Lagrange equation for a functional that doesnt depend on x I get
$$F-y'\frac{\partial F}{\partial y'}=c$$
$$\Leftrightarrow y^2(1-(y')^2)=c$$
$$\Leftrightarrow \int \frac{1}{\sqrt{1-\frac{c}{y^2}}}dy=\int dx$$
$$\Leftrightarrow y=\frac{-c}{x^2-1}$$
Now I have to sub this y(x) into the original integral and I am comfortable doing the integral apart from what to do for the upper and lower limits of integration.

2. Oct 28, 2014

### Staff: Mentor

There should be another free parameter from the integration.
You can use the limits of the integration to fix those two parameters.

3. Oct 28, 2014

### Dick

That doesn't look like any version of Euler Lagrange I know. I'd try looking it up again. You should get a 2nd order ODE.

4. Oct 28, 2014

### jimmycricket

In this case F does not depend on x so the E-L equation is reduced to
$$F-y'\frac{\partial F}{\partial y'}=c$$ and you are left with a 1st order ode.