Euler-Largrange equation is
Please calculate this
Using the Weierstass condition, find the strongly minimizing curve and the value of for the cases
where .
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.
Let
Using the Euler-Largrange eqn, I get
so ...(1)
which is a Euler's type DE which can be solved by changing variable .
Let . By the chain rule and noting that :
........(2)
and
....(3)
Subtracting (2) and (3) I get,
implying
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.
Thank-you.
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?