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https://en.wikipedia.org/wiki/Brachistochrone_curve

In this article, it points out that the curve taking the shortest time from point A to point B under constant acceleration has a different segment from the Tautochrone curve.

How come the minimal time isn't taken from the top of the curve to its vertex? Why is this extra upward-sloping component necessary to minimize travel time past the vertex?

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  • $\begingroup$ By allowing the bead to fall farther, it moves faster (that is, more potential energy is converted to kinetic) $\endgroup$
    – Digiproc
    Commented Jan 8, 2021 at 15:22
  • $\begingroup$ Can you add a bit more details to your question to clarify what you mean? $\endgroup$
    – Lost
    Commented Jan 8, 2021 at 15:55
  • $\begingroup$ The curve comes from making it a solution of Euler-Lagrange equation. If you need a physical explantion then add more details to the question and also maybe look at geometric solutions and such others which are mentioned on the wikipedia as well. $\endgroup$
    – Lost
    Commented Jan 8, 2021 at 15:57
  • $\begingroup$ I think Digiproc answered it, but thank you. $\endgroup$
    – PhiEarl
    Commented Jan 8, 2021 at 20:21

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