I'm trying to find the soltutionsolution of the brachystochronebrachistochrone problem. I'm was wondering if there is a way to solve it directly using Mathematica. The problem is the following:
$$J[t]=\frac1{\sqrt 2 g}\int_{y_1}^{y_2}\frac{\sqrt{1+y^{\prime 2}}}{\sqrt y}\,dy$$
I have to find the equation y(x)$y(x)$ that minimizes the functional J[t]$J[t]$. This functional (y(x)$y(x)$) is the path of minimum time that a body will move from a point y1$y_1$ to a point y2$y_2$.
To achieve that, it is needed to use the Euler equation:
$$\frac{df}{dy}-\frac{d}{dx}\left(\frac{df}{dy^\prime}\right)=0$$
where
$$f=\frac{\sqrt{1+y^{\prime 2}}}{\sqrt y}$$
The code below is the solution of the Euler equation that I was trying to integrate without sucesssuccess.
Integrate[(-1 - y'[x]^2 - 2 y[x] y''[x])/(
2 y[x]^(3/2) (1 + y'[x]^2)^(3/2)), {y[x], 0, 1}]
This code returns:
Integral of
(-1-(y^\[Prime])[x]^2-2 Integrate`$$a$201005 (y^\[Prime]\[Prime])[x])/Integrate`$$a$201005^(3/2)does not converge on{0,1}. >>
