- #1
Erbil
- 57
- 0
Create the Euler-Lagrange equation for the following questions (if it's necessary change the variables).
$$\tag{1}\int _{y_{1}}^{y2}\dfrac {x^{'}{2}} {\sqrt {x^{'}{2}+x^{2}}}dy$$
$$\tag{2}\int _{x_{1}}^{x_{2}}y^{3/2}ds $$
$$\tag{3} \int \dfrac {y.y'} {1+yy{'}}dx $$
$$ \dfrac {d} {dy}\left( \dfrac {\partial F} {\partial x^{'}}\right) -\dfrac {\partial F} {\partial x}=0 $$
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.
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.