Euler - Lagrange Equation(changing variable)

[Erbil]

Create the Euler-Lagrange equation for the following questions (if it's necessary change the variables).

Homework Statement

$$\tag{1}\int _{y_{1}}^{y2}\dfrac {x^{'}{2}} {\sqrt {x^{'}{2}+x^{2}}}dy$$

$$\tag{2}\int _{x_{1}}^{x_{2}}y^{3/2}ds $$

$$\tag{3} \int \dfrac {y.y'} {1+yy{'}}dx $$

Homework Equations

$$ \dfrac {d} {dy}\left( \dfrac {\partial F} {\partial x^{'}}\right) -\dfrac {\partial F} {\partial x}=0 $$

The Attempt at a Solution

I don't have an idea about 1 and 3.But here it's what I have tried for 2.

2) $$\int _{x_{1}}^{x_{2}}y^{3/2}ds = \int _{x_{1}}^{x_{2}}y^{3/2}(1+y'^{2})^{1/2}= \int _{y_{1}}^{y2}\ (1+x^{'2}) (y^{3/2})dy $$
So our Euler equation is;
$$ \dfrac {d} {dy}\left( \dfrac {\partial F} {\partial x^{'}}\right) -\dfrac {\partial F} {\partial x}=0 $$

*Then I have to find Y' or X'.But I did not take a differential equations course yet,we use Beltrami identity to calculate the extremum points.

 

[haruspex]

here it's what I have tried for 2.

2) $$\int _{x_{1}}^{x_{2}}y^{3/2}ds = \int _{x_{1}}^{x_{2}}y^{3/2}(1+y'^{2})^{1/2}= \int _{y_{1}}^{y2}\ (1+x^{'2}) (y^{3/2})dy $$

You dropped a sqrt: $$F(y, x, x') = (1+x'^2) ^\frac12 y^\frac32 $$

So our Euler equation is;
$$ \dfrac {d} {dy}\left( \dfrac {\partial F} {\partial x^{'}}\right) -\dfrac {\partial F} {\partial x}=0 $$

So what do you get when you substitute for F?

 

[Erbil]

http://math.stackexchange.com/quest...ange-equation-changing-variable/415427#415427

He is saying that changing the variables in this situation is immaterial.

 

[Erbil]

You dropped a sqrt: $$F(y, x, x') = (1+x'^2) ^\frac12 y^\frac32 $$

So what do you get when you substitute for F?

I think it will be ;

$$F(y, x, x') = (1+x'^2)/x' ^\frac12 y^\frac32 $$

 

[haruspex]

I think it will be ;

$$F(y, x, x') = (1+x'^2)/x' ^\frac12 y^\frac32 $$

How do you get that? Isn't F the function in the integral wrt y?

 

[Erbil]

How do you get that? Isn't F the function in the integral wrt y?

Before that,I would like to ask can I use the method what I've linked for solve these problems?

 

[haruspex]

Before that,I would like to ask can I use the method what I've linked for solve these problems?

Looks ok to me. It does get confusing the way you switch between the ∫f(x, y, y')dx form and the ∫f(y, x, x')dy form. Pls fix on one.