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Euler Lagrange Equation : if y(x) is a curve which minimizes/maximizes the functional :

F[tex]\left[y(x)\right][/tex] = [tex]\int^{a}_{b} [/tex]f(x,y(x),y'(x))dx

then, the following Euler Lagrange Differential Equation is true.

[tex]\frac{\partial}{\partial x}[/tex] - [tex]\frac{d}{dx}(\frac{\partial f}{\partial y'})[/tex]=0

Well...

I don't understand why the function f has only three variables x, y(x) and the derivative of that.

what about y'' or y[tex]^{(3)}[/tex]? I think it could be possible.(physically) All files related to this topic states that the function f as a function of variables x, f(x), and f'(x).

i.e. can function f be like : f(x,y(x),y'(x),y''(x),...) ? or... is it unnecessary to think about the second derivative and furthermore?

F[tex]\left[y(x)\right][/tex] = [tex]\int^{a}_{b} [/tex]f(x,y(x),y'(x))dx

then, the following Euler Lagrange Differential Equation is true.

[tex]\frac{\partial}{\partial x}[/tex] - [tex]\frac{d}{dx}(\frac{\partial f}{\partial y'})[/tex]=0

Well...

I don't understand why the function f has only three variables x, y(x) and the derivative of that.

what about y'' or y[tex]^{(3)}[/tex]? I think it could be possible.(physically) All files related to this topic states that the function f as a function of variables x, f(x), and f'(x).

i.e. can function f be like : f(x,y(x),y'(x),y''(x),...) ? or... is it unnecessary to think about the second derivative and furthermore?

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