Just starting a course on Lagrangian Mechanics and I'm just wondering what about the Euler-Lagrange equation, and more specifically what I'm meant to be trying to do ..
One of the questions from my textbook reads :
Solve the Euler-Lagrange equation for the following function \begin{align*} f(y,y') = y^2+y'^2 \end{align*} Looks simple enough... But where should I be headed to start?
The Euler-Lagrange Equation as we have it is \begin{align*} \frac{\partial f}{\partial y} = \frac{d}{dx} \frac{\partial f}{\partial y'} \end{align*}
Do I just find the partial derivatives of f treating y and y' as independent variables as follows:
$\frac{\partial f}{\partial y} = 2y,$ $\frac{\partial f}{\partial y'} = 2y$ so then we have : $2y = \frac{d}{dx}(2y')$ $y = \frac{d}{dx}(y')$ $yx+c = y'$ $\frac{yx^2}{2}+cx+d = y$
And then simplify with the y isolated on the right so it looks nice .. or at least thats what I thought..
Any comments ? right track, wrong track? Also, what is this euler equation actually solving for? Is it just a nice way of solving differential equations in this form ?
When we did this in class, we did a monster proof showing it minimizes the path between 2 points or finds the extrema for the function $I(x) = \int^a_b \ F(y(x),y'(x),x) dx,$ which is all great but I'm not too sure how this relates to the equation we solve in the question .. Thanks for all