1. ## Calculus of Variations.

I’m practicing some problems for applied mathematics, and couldn’t quite solve the following:
In calculus of variations, if the speed is x, and the travel time is t = Integral(0 to 1) (1/x)Sqrt(1+(u’)²)dx with u(0)=0 and u(1)=1, how can see from the Euler equation that u’/x(sqrt(1+(u’)²)) is constant?
And how can I integrate once more in order to get the optimal path u(x)?
If you know the solution, or could guide me in the right direction, I would appreciate it a lot! Thanks in advance..

2. Originally Posted by francoiskalff
I’m practicing some problems for applied mathematics, and couldn’t quite solve the following:
In calculus of variations, if the speed is x, and the travel time is t = Integral(0 to 1) (1/x)Sqrt(1+(u’)²)dx with u(0)=0 and u(1)=1, how can see from the Euler equation that u’/x(sqrt(1+(u’)²)) is constant?
And how can I integrate once more in order to get the optimal path u(x)?
If you know the solution, or could guide me in the right direction, I would appreciate it a lot! Thanks in advance..
If $\frac{u'}{x\sqrt{1+u'^2}} = c_1$

Square both sides and solve for $u'^2$ - then solve for $u'$.

3. Originally Posted by Danny
If $\frac{u'}{x\sqrt{1+u'^2}} = c_1$

Square both sides and solve for $u'^2$ - then solve for $u'$.
Hey thanks for your reply, but I think you misinterpreted my question. I guess I wasn't clear enough. I meant to ask:
if the speed is x, and the travel time is t = Integral(0 to 1) (1/x)Sqrt(1+(u’)²)dx with u(0)=0 and u(1)=1, how can you see from the Euler equation WHICH QUANTITITY is constant (Snell's law)?
$\frac{u'}{x\sqrt{1+u'^2}} = c_1$ but i dont know how to get there...