Euler-Largrange equation is
Please calculate this
Using the Weierstass condition, find the strongly minimizing curve and the value of for the cases
I've gotten to a point where I have tried to continue, but I can't seem to get anything that resembles a correct solution.
Using the Euler-Largrange eqn, I get
which is a Euler's type DE which can be solved by changing variable .
Let . By the chain rule and noting that :
Subtracting (2) and (3) I get,
which is equivalent to equation (1).
The extremal is therefore
Now, when using I don't get a solution that appears to be correct.....and I don't know how to continue. Any help would be greatly appreciated.
As zzzoak mentioned, you must compute
Now, you're viewing which means that the derivative in question must be computed using the product rule (it's a total derivative, not a partial derivative!). I get
Another puzzle: you're asked to use the Weierstrass condition. Is that the Weierstrass-Erdmann Corner Condition?
I messed that up....twice.
Now I have
There is a hint to this question that I forgot to mention earlier, which says "Look for solutions of the Euler-Lagrange equation of the form .
That doesn't really make sense to me...how can I get the extremal from here?