# Euler - Lagrange Equation(changing variable)

1. Jun 1, 2013

### Erbil

Create the Euler-Lagrange equation for the following questions (if it's necessary change the variables).
1. The problem statement, all variables and given/known data

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

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

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

2. Relevant equations

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

3. 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.

2. Jun 2, 2013

### haruspex

You dropped a sqrt: $$F(y, x, x') = (1+x'^2) ^\frac12 y^\frac32$$
So what do you get when you substitute for F?

3. Jun 17, 2013

### Erbil

4. Jun 17, 2013

### Erbil

I think it will be ;

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

5. Jun 18, 2013

### haruspex

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

6. Jun 18, 2013

### Erbil

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

7. Jun 18, 2013

### haruspex

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.