I am a bit confused by the following problem, I am comfortable with just one Euler-Lagrange (I think!) equation but this has two and has me a bit confused :

Solve the following using Euler-Lagrange equations

$\displaystyle J[\vec{x}]=\int_{a}^{b}\frac{\sqrt{(x'_1)^2+(x'_2)^2}}{x_2}d t$

Thanks for any help

(Is there any way to get LaTex anymore??)

n

So this is what I have so far (didn't post earlier as I couldn't get the LaTex to work)

The Two Euler-Lagrange equations give:

1. $\displaystyle 0-\frac{d}{dt}(\frac{x'_1}{x_2\sqrt{(x'_1)^2+(x'_2)^ 2}})=0$

2.$\displaystyle -\frac{\sqrt{(x'_1)^2+(x'_2)^2}}{x_2^2}-\frac{d}{dt}(\frac{x'_2}{x_2\sqrt{(x'_1)^2+(x'_2)^ 2}})=0$

So we may conclude that $\displaystyle \frac{x'_1}{x_2\sqrt{(x'_1)^2+(x'_2)^2}}=k $ and so

$\displaystyle \frac{x'_1}{x_2*k}=\sqrt{(x'_1)^2+(x'_2)^2}$

substituting this into equation 2. we get

$\displaystyle -\frac{x'_1}{x^3_2*k}-\frac{d}{dt}(\frac{x'_2}{x_1'})$

However I have no idea what to do now