# Thread: Finding Extremal - Stuck in the middle

1. ## Finding Extremal - Stuck in the middle

I have a Euler-Lagrange equation

$\displaystyle \frac{\partial f}{\partial x} - \frac{d}{dt} (\frac{\partial f}{\partial \dot{x}})= 0$

which for my f is

$\displaystyle \dot{x} + 1 - \frac{d}{dt} (\dot{x} + x + 1) = 0$

The solutions say I should end up with

$\displaystyle \ddot{x} - 1 = 0$

I end up with

$\displaystyle \dot{x} + 1 - \ddot{x} - \dot{x} - x = 0$

$\displaystyle \ddot{x} + x = -1$

I must be doing

$\displaystyle \frac{d}{dt} (\dot{x} + x + 1)$

incorrectly? What am I doing wrong?

2. ## Re: Error

Yes, you did appear to differentiate incorrectly. Noting that,
$\displaystyle \frac{d}{dt}(\dot{x}+x+1)=\frac{d}{dt}\dot{x}+\fra c{d}{dt}x+\frac{d}{dt}1$,
I'd ask you what is $\displaystyle \frac{d}{dt}1$?

--Kevin C.

3. Yes, I ended up figuring out where I went wrong. I am stuck on this one however....

$\displaystyle \frac{d}{dt} (1 + 2t^2 \dot{x}) = 0$

I get

$\displaystyle 2t^2 \dot{x} = C$

$\displaystyle \dot{x} = \frac{1}{2} C t^{-2}$ ......wondering if $\displaystyle \frac{1}{2}$ should be absorbed by the constant

$\displaystyle x = -C t^{-3} + B$

but the solution is

$\displaystyle x = -At^{-1} + B$

4. You're integrating incorrectly. The exponent gets more positive by one when you integrate. It does not become more negative.

5. Cannot believe I missed that.

6. I've seen loads of people do that sort of thing. They're studying something more advanced, and they are so focused on learning the more advanced thing that they forget some elementary principle. Ah, well. Have a good one!