Derivation of the Brachistochrone | Euler-Lagrange Equation

We derive the explicit form x(y) of the brachistochrone, the curve of fastest descent.

[emilianogdeurreta]

Hello, I wanted to know how to set up the initial conditions to find the integration constants. It's clear that x(y = 0) = 0, since the motion starts at the point (0, 0). However, what I can't figure out is the condition for x'(y = 0), because what we know is that dot{x} (t = 0) = 0. But when I try to translate this to x'(y = 0), I end up with dot{x(y(t))} = x'(y)dot{y}(t). At t = 0, both dot{x} and dot{y} are zero, so by this approach, I can't obtain a condition on x'(y = 0). What am I doing wrong?
Translation done. I hope it's right.

[emilianogdeurreta]

Hola. Quería saber cómo plantear las condiciones iniciales para hallar las constantes de integración. Es claro que x(y = 0) = 0, ya que el movimiento comienza en el punto (0, 0). Lo que no puedo plantear sería la condición x'(y = 0), ya que lo que sabemos es que \dot{x} (t = 0) = 0, pero al tratar de traducirlo a x´(y = 0), me queda que \dot{x(y(t))} = x´(y)\dot{y}(t), y para t = 0, tanto \dot{x} como \dot{y} son cero, así que por este camino no puedo obtener alguna condición sobre x'(y = 0) ¿Qué estoy haciendo mal? Muchas gracias. PD: Perdón por no escribirlo en inglés. Espero que lo puedas traducir y entender. Ahora veo de tratar de traducirlo.

[OrgStinx]

Thanks, this is awesome. Btw, I don't like derivation using Fermat's principle, because it's illogical. Why the heck light has mind? It's just a wave.

[mrslave]

6:38 "i'm going to treat y as the independent variable" - you can't. the way you drew the path of the solution, there are 2 x values. this is a violation of the definition of a function. x cannot be a function of y. y cannot be treated as the independent variable. in order to fix this issue, you need to take the +/- of the square root (whenever you have one). for example cy=sin^2θ => +/- sqr(cy)=sineθ.