Variational problems in two variables

Sep 2006
782
100
The raggedy edge.
Write down the Euler-Lagrane equations for critical points of the functional I of two functions \(\displaystyle x(t), \ y(t)\) where:

\(\displaystyle I(x,y)=\int_{t_1}^{t_2}(x^2+y^2)\sqrt{1+(x')^2+(y')^2}\)
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 \(\displaystyle f(x,y,y',x')=(x^2+y^2)\sqrt{1+(x')^2+(y')^2}\) so my first equation is:

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

The second equation would be the same but with \(\displaystyle y'\) instead of \(\displaystyle x'\).

Am I reading this correctly?
 

Jester

MHF Helper
Dec 2008
2,470
1,255
Conway AR
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 \(\displaystyle f(x,y,y',x')=(x^2+y^2)\sqrt{1+(x')^2+(y')^2}\) so my first equation is:

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

The second equation would be the same but with \(\displaystyle y'\) instead of \(\displaystyle x'\).

Am I reading this correctly?
It might be clearer if we write it as

\(\displaystyle
\frac{\partial L}{\partial x} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) = 0
\)

\(\displaystyle
\frac{\partial L}{\partial y} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{y}}\right) = 0.
\)