Find the extremals of with and subject to the constraint
Okay, first of all, could you please what extremals are in a physical sense? I don't quite understand that.
Anyway, I haven't got very far, but here is what I have.
Since is independent of , it follows from the Euler-Lagrange equation that
so
Therefore
I don't really know where to go from here.
You're doing calculus of variations here. Calculus of variations is all about finding functions that maximize or minimize a functional. A functional, in this case, maps functions to numbers. Your functional in this problem is
So you have to find an extremum, or extremal, (doesn't matter if it's a min or a max) of this functional over the collection of functions such that
and
I don't think your application of the Euler-Lagrange equations is correct. Moreover, in order to satisfy the integral constraint of
I think you're going to have to use the method of Lagrange multipliers. You should incorporate the Lagrange multiplier before you apply the Euler-Lagrange equation.
Does any of this ring a bell?
Yes, when I originally posted this we hadn't done it in class yet and I was trying to get ahead. Now I have something......
Define
and
The Lagrangian is
We can apply the Euler-Lagrange equation to obtain
and therefore
So
(where c is a constant and k is a constant)
Then from we have
using
Using the constraint ,
and here is where I am having trouble. I don't think the next line is correct.
I don't think you're applying the B.C. x(2) = 1 correctly. Everything looks good up until that point. What I would do is this: once you've applied the B.C. x(0) = 0, you find out that C = k. So re-write x(t) using the value for k. Then apply the second boundary condition. That'll give you an equation relating lambda to C (assuming you've eliminated k - you could just as easily eliminate C. It will make no difference in the final answer.) Then apply the constraint equation. And no, I don't think you're applying the constraint equation correctly, either. Check that again, but with using the method I've outlined.
You've still got a sign error in applying x(2) = 1. So here's how you get good at algebra: pretend someone is holding a gun to your head, and if you make a mistake, that someone is going to pull the trigger! Or, you can just work in a high-energy physics lab. With those 10,000 volt wires running around, if you make a mistake, you're dead.
Imaginary gun to my head.....check.
Working in a high-energy physics lab with 10,000 volt wires running around.....check.
Standing over a tank full of ill-tempered mutated see Bass and sharks with frickin laser-beams attached to their head for good measure....check.
Working through carefully this time I find the extremal to be
which at and at
I'm alive!