Weierstass's Excess Function - minimizing curve
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.