# Variational problems in two variables

#### Showcase_22

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 $$\displaystyle x'$$.

#### Jester

MHF Helper
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 $$\displaystyle x'$$.

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