# Euler-Lagrange Single function of single variable with higher derivatives

Here is the page on Wikipedia:

So it says the fixed boundary conditions for the function itself as well as for the first $n-1$ derivatives. You can fix the boundary points physically say $y(a)=a'$ and $y(b)=b'$ but what about for the derivatives? For example for this problem from my lectures-

We've fixed $y(0) =0$ for this diving board problem - where does $y'(0)=0$ come from?

Then in the variant where the swimmer holds the board up we fix the the other end of the board $x=L$ at a certain height so $y(L)=$ "whatever". We need 4 boundary condtions here where is the other? I assume by Wikipedia article that $y'(L)$ is something- is the other fixed boundary condtion but what is it- it's not mentioned in the notes...

Would appreciate any clarification of the ambiguity thanks...

In beam theory, beam is modelled as line curve with some stiffness properties. As such in 2D it can be modelled as a continously differentiable curve. Usually a fixed end is made as a horizontal groove in the fixed wall whereby the curve (i.e. the beam) will remain tangent at the location of the horizontal groove. Therefore, $y'(0)=0$
The swimmer's arm can be taken a simple support which can't provide an external moment. Hence the bending moment at the other end must also be zero. Hence by dynamic considerations $y''(L) = 0$. Thus you have the four boundary conditions viz. $y(0) =0, y'(0)=0, y(L) =$"whatever"$, y''(L) = 0$.