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..
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)?
I know the answer is:
but i dont know how to get there...
Again, the same question posted in several forums.
(But the OP did not reply so far, whether he finds some of the answers satisfactory.)
Google Groups
francoiskalff variations - Google Search
http://www.mathhelpforum.com/math-he...ariations.html
http://www.mymathforum.com/viewtopic.php?f=27&t=13617
S.O.S. Mathematics CyberBoard :: View topic - Calculus of Variations