Variational problems in two variables

**Showcase_22** · May 17th 2010, 02:33 AM

Write down the Euler-Lagrane equations for critical points of the functional I of two functions $x(t), \ y(t)$ where:

$I(x,y)=\int_{t_1}^{t_2}(x^2+y^2)\sqrt{1+(x')^2+(y' )^2}$

I'm slowly working my way through this sheet, but I keep encountering things not in the notes. I'm also having trouble understanding what wikipedia has to say: Euler?Lagrange equation - Wikipedia, the free encyclopedia

I have a function $f(x,y,y',x')=(x^2+y^2)\sqrt{1+(x')^2+(y')^2}$ so my first equation is:

$\frac{\partial f}{ \partial y}-\frac{\partial}{\partial x} \left( \frac{\partial f}{\frac{\partial y}{\partial x}} \right)- \frac{\partial}{x'} \left( \frac{\partial f}{\frac{\partial y}{\partial x'}} \right)$

The second equation would be the same but with $y'$ instead of $x'$ .

Am I reading this correctly?

**Danny** · May 17th 2010, 06:10 AM

Originally Posted by Showcase_22

I'm slowly working my way through this sheet, but I keep encountering things not in the notes. I'm also having trouble understanding what wikipedia has to say: Euler?Lagrange equation - Wikipedia, the free encyclopedia

I have a function $f(x,y,y',x')=(x^2+y^2)\sqrt{1+(x')^2+(y')^2}$ so my first equation is:

$\frac{\partial f}{ \partial y}-\frac{\partial}{\partial x} \left( \frac{\partial f}{\frac{\partial y}{\partial x}} \right)- \frac{\partial}{x'} \left( \frac{\partial f}{\frac{\partial y}{\partial x'}} \right)$

The second equation would be the same but with $y'$ instead of $x'$ .

Am I reading this correctly?

It might be clearer if we write it as

$<br /> \frac{\partial L}{\partial x} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) = 0<br />$

$<br /> \frac{\partial L}{\partial y} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{y}}\right) = 0.<br />$

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