- #1
Pablo Brubeck
- 7
- 0
I attempt to solve the brachistochrone problem numerically. I am using a direct method which considers the curve ##y(x)## as a Lagrange polynomial evaluated at fixed nodes ##x_i##, and the time functional as a multivariate function of the ##y_i##. The classical statement of the problem requires the curve ##y(x)## to have its endpoints on ##(x_0, 0)## and ##(x_1, y_1)## and the initial velocity of the particle to be ##v_0=0##, so that the time functional would take the form
$$ T[y]=\int_{x_0}^{x_1} \sqrt{\frac{1+\left[\frac{dy}{dx}\right]^2}{2gy}} dx $$.
The derivative ##\frac{dy}{dx}## is approximated from the Lagrange polynomial and the integral is computed using a quadrature rule, then the functional is minimized using an interior point method. But the problem comes when the denominator of the integrand vanishes, which will always happen at the endpoint ##(x_0, 0)##. I tried to avoid this by setting ##v_0##, but the method fails to converge to a continuous curve for arbitrary small ##v_0##.
What would be the proper way to handle this situation?
$$ T[y]=\int_{x_0}^{x_1} \sqrt{\frac{1+\left[\frac{dy}{dx}\right]^2}{2gy}} dx $$.
The derivative ##\frac{dy}{dx}## is approximated from the Lagrange polynomial and the integral is computed using a quadrature rule, then the functional is minimized using an interior point method. But the problem comes when the denominator of the integrand vanishes, which will always happen at the endpoint ##(x_0, 0)##. I tried to avoid this by setting ##v_0##, but the method fails to converge to a continuous curve for arbitrary small ##v_0##.
What would be the proper way to handle this situation?