Euler-Lagrange equations

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June 30th 2010, 02:42 AM
punkstart

Euler-Lagrange equations

The question reads " Find the extremal of class $C^2$ ,if it exists,which satisfies the given final conditions for problems with lagrangian as follows:
$L(x,y,\dot{x},\dot{y})=\dot{x}^2+\dot{y}^2+2xy$ ;
$x(0)=y(0)=0$
$x(\frac{\pi}{2})=y(\frac{\pi}{2})=1$

Please check my work:

The Euler-Lagrange Equation given by $\frac{d}{dt}(\frac{\partial L}{\partial \dot{x}})-\frac{\partial L}{\partial x} = 0$

I get (for x) $\frac{\partial L}{\partial \dot{x}} = 2\dot{x}$

$\frac{d}{dt}2\dot{x}=2\ddot{x}$

$\frac{\partial L}{\partial x}=2y$

Which yields $2\ddot{x}-2y=0$

integrating wrt t: $\int 2\ddot{x} dt = \int 2y dt$
$2\dot{x}=2yt+c$
$\int 2\dot{x} dt= \int (2yt+c) dt$
so $2x=yt^2 +ct + d$
So that i have for x, $x=\frac{1}{2}[yt^2+ct+d]$
$x(0) => d=0; x(\frac{\pi}{2}) => c=\frac{2}{\pi}(1-\frac{\pi^2y}{8})$ didn't show calc's but i'm certain here

I find then that $x=\frac{1}{2}yt^2+\frac{2}{\pi}(1-\frac{\pi^2y}{8})t$
And for y, $y=\frac{1}{2}xt^2+\frac{2}{\pi}(1-\frac{\pi^2x}{8})t$

Am i done at this point? Have i found an extremal of the lagrangian?
I am not sure of the differentiation and integration ? I don't have much in the way of examples to work with...just started studying these...

Another question, what happens if the equation is trivially zero? i.e $\frac{d}{dt}(\frac{\partial L}{\partial \dot{x}}) = \frac{\partial L}{\partial x}$ ?
Does this mean that the lagrangian will have no extremal at all? Or is there some operation to be carried out, or manipulation to get an extremum? What must happen for there to be no extremum to the lagrangian?

One more thing, can anyone recommend a good textbook that deals with variational calculus comprehensively, with examples and solutions to exercises ? I only have a study guide,which is a bit of a cut and paste half story of the subject.

THANK YOU !
June 30th 2010, 03:59 AM
Ackbeet

I don't think your E-L equations are correct. You've got $L=\dot{x}^{2}+\dot{y}^{2}+2xy$ .

You will need two E-L equations, one for each variable, although by symmetry, you can easily tell that the x and y equations must be the same.

Go step by step!

$\frac{\partial L}{\partial\dot{x}}=2\dot{x}$

$\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}=2\ddot{x}$

$\frac{\partial L}{\partial x}=2y$ .

So your E-L equation looks like

$2\ddot{x}=2y$ , or $\ddot{x}=y.$

For the y E-L equation, you're going to get

$\ddot{y}=x.$

Can you solve from here?
June 30th 2010, 04:04 AM
Ackbeet

As for a good book on Calculus of Variations, I would go with Variational Calculus and Optimal Control: Optimization with Elementary Convexity, by John Troutman.

Incidentally, while this post is Calculus of Variations, I would probably post it in Advanced Applied Math.
July 1st 2010, 12:54 AM
punkstart

Thanks Ackbeet, all i know of what an extremal is, is that it is a solution to the euler lagrange equations. I don't think i can represent x and y any simpler here, so i think i have what i need.
July 1st 2010, 02:57 AM
Ackbeet

You're very welcome. Have a good one!