# Brachistochrone problem for 3 points

I wonder how I can solve the Brachistochrone problem for 3 points? The matter starts from point A that is the highest point and it must pass from B and must finish with point C. (No any friction in the system) I wonder what the shortest time path for that problem?

Is it enough to use Johann Bernoulli's solution from A to B and then Use the solution from B to C?

Or do I need to follow a different way to solve the shortest time path problem for 3 points?

Yes, but one first has to generalize the classical 2-point Brachistochrone problem $A \to B$ where the initial speed $v_A$ traditionally is zero, to the case where the initial speed $v_A$ may be non-zero but fixed. The solution to this initial speed Brachistochrone problem (assuming no friction) is still a cycloid.
Now consider the 3-points Brachistochrone problem $A \to B\to C$ with initial speed $v_A$. The speed $v_B$ is given by energy conservation alone. Thus the two segments $A \to B$ and $B \to C$ are completely decoupled, and they can be optimized as two independent 2-point Brachistochrone problems with initial speeds $v_A$ and $v_B$, respectively, leading to two corresponding cycloids $A \to B$ and $B \to C$.
• Notes for later: $1+ (\frac{dx}{dy})^2=\frac{k^2}{y}$. Rescale $x$ and $y$ so that $k^2=2$. Then $\frac{dx}{dy}= +(\frac{2}{y}-1)^{-1/2}=\sqrt{\frac{y}{2-y}}$ with antiderivative $x=\arctan \frac{\sqrt{y(2-y)}}{1-y}-\sqrt{y(2-y)}=\theta-\sin\theta$, where $1-y=\cos\theta$. Aug 2 '19 at 9:06