What is Brachistochrone: Definition and 36 Discussions

In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696.
The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can only use up to the first half rotation, and always ends at the horizontal. The problem can be solved using tools from the calculus of variations and optimal control.The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B. If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time will differ from the tautochrone curve.

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  1. D

    B Verifying AI Chatbot's Answer on Brachistochrone Curve Formula

    I was interviewing an AI chatbot and was wondering how good it is at physics… can anyone confirm whether the answer it gave is true or false? This was through a chatbot called ChatGPT. My prompt: What’s the formula for the Brachistochrone curve? The AI answer: The Brachistochrone curve is a...
  2. D

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    Hello, There is a physics problem called the Brachistochrone problem which I know has been solved for 0 initial velocity (assumes 0 friction and only gravity) and I know the answer is a cycloid. My question is: is there is an existing formula for finding the portion of a cycloid which is the...
  3. physicsbeginnerss

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    This is 'Boas mathematical Methods in the Physical Sciences' homework p484.(Calculus of Variations) problem2 section4 number 2 The bead is rolling on the cycloid curve.(Figure 4.4) And the book explain that 'Then if the right-hand endpoint is (x, y) and the origin is the left-hand endpoint...
  4. physics_cosmos

    I Solving the Brachistochrone problem with friction

    This Wolfram Alpha Page contains a derivation of the parametric form of the brachistochrone curve that result from either assuming friction or its absence. I am asking for help understanding how the solution to the differential equation obtained from applying the Euler-Lagrange equation to the...
  5. The proof of Snell’s law using springs - 3blue1brown

    The proof of Snell’s law using springs - 3blue1brown

    The law of light bending in matter. Also an important law for the proof of brachistochrone problem.
    • YoungPhysicist
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    • brachistochrone refraction snell's law
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    • Category: Optics
  6. A

    I Solution to Brachistochrone Problem

    Can anybody post a full solution of the Brachistochrone problem provided by Newton (with full explanations) ? Or, any source about the same would be much helpful. Thank you in advance !
  7. J

    Brachistochrone problem with friction

    Hey, I am doing some research on the brachistochrone problem WITH frictions. I found the following demonstration on the web. The beginning is ok. But I can't understand how the managed to pass between (29) and (30) and between (30) and ((32), (33)). If someone could help me, it would be very...
  8. Joe Wolf

    I Brachistochrone for Specific Ratios

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  9. Z

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  10. zwierz

    I History of brachistochrone and catenary

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  11. J

    What are some ways to further explore the brachistochrone problem?

    Homework Statement Hi I'm in second year of study in Math, Physic and Informatic and I require some help. I began a work on the brachistochrone problem. It's really interesting and I already found lot of things (the equation of the cycloid by the Bernoulli's method, I wrote some programs which...
  12. F

    B The Brachistochrone Curve - Which theories apply?

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  13. wrobel

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  14. Hamza Abbasi

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  16. G

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  17. D

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  18. B

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  19. L

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  20. J

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  21. W

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  22. H

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  23. G

    Brachistochrone with velocity - still a cycloid

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  24. N

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  25. J

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  26. 9

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  27. T

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  28. T

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  29. T

    N.Solve Brachistochrone Homework: Find Time T to Slide Down from Height 0 to -h

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  30. Z

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  31. E

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  32. M

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  33. S

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  34. S

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  35. H

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  36. E

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