What is Brachistochrone problem: Definition and 17 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

    Brachistochrone Problem w/ Initial Velocity

    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...
  2. physicsbeginnerss

    The Brachistochrone Problem: cycloid curve

    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...
  3. 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...
  4. The brachistochrone problem

    The brachistochrone problem

    Brachistochrone problem’s original proof by Snell’s law and a story of Newton.
  5. 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 !
  6. 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...
  7. 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...
  8. wrobel

    Insights General Brachistochrone Problem - Comments

    wrobel submitted a new PF Insights post General Brachistochrone Problem Continue reading the Original PF Insights Post.
  9. A

    Bead sliding on a wire - calculus of variations

    I am asked to find the shape of a wire that will maximize the speed a sliding bead when it reaches the end point(Similar to the brachistochrone problem expect that the speed is to be maximized and not time minimized). But shouldn't the speed at the end be independent of the shape of the wire...
  10. Hamza Abbasi

    Brachistochrone Equation Problem

    Homework Statement This is the solution of Brachistochrone . Homework EquationsThe Attempt at a Solution I am very confused that how the x in equation(6.24) get its value a(1-cosӨ) ? What is the technique behind this solution of x?
  11. J

    Euler-Lagrange Brachistochrone Problem in rotating system

    Homework Statement Bead slides on a wire (no friction) shaped as r = r(\theta) in the Oxy plane. The Oxy plane and the constraining wire rotate about Oz with \omega = const r, \theta is the rotating polar frame; r, \phi is the stationary frame. Find the trajectory r = r(\phi) in the...
  12. W

    Minimizing the Functional for the Brachistochrone Problem

    Homework Statement So if +x points downward and +y points rightwards then the functional that needs to be minimized is: \sqrt{2g}T[y]=\int_{x_0}^{x_1}\frac{dx}{\sqrt{x}}\sqrt{1+\left(\frac{dy}{dx}\right)^2} Homework Equations I think we just have to use the Euler lagrange...
  13. J

    ODE - Brachistochrone Problem

    y[1+(y')^2] = k First solve for dx in terms of y and dy, an then use the substitution y = ksin2(θ) to obtain a parametric form of the solution. The curve turns out to be a cycloid. My attempt: (y')^2 = k/y-1 dy/dx = sqrt(k/y-1) dx = dy/[sqrt(k/y-1)] then substitute y =...
  14. 9

    How do I solve the Brachistochrone Problem with a given differential equation?

    Hello, I'm having problems with a D.E. question, I'm asked to solve the equation: \left(1+y^{'2}\right)y=k^{2} where K is a certain positive integer to be determined later. This more commonly known, as you probably know, as one of the solutions to the Brachistochrone Problem. I really...
  15. S

    How can the Brachistochrone problem be solved using parametric equations?

    find the curve for which the body will follow such that the time of travel is a minimim. Hints Minimize t_{12} = \int_{x_{1}}^{x_{2}} dt = \int_{x_{1}}^{x_{2}} \frac{ds}{v} = \int_{x_{1}}^{x_{2}} \sqrt{\frac{1+y'^2}{2gy}} dx since F does not depend on x i can use hte beltrami identity (from...
  16. S

    Precursor to brachistochrone problem

    Another long question but not that hard. Most of the writing is my work/questions According to my prof if i cna solve this... the resulting relation can be used to solve Bernoulli's problem For \delta \int_{x_{1}}^{x_{2}} F(x,y(x),\dot{y}(x)) dx = 0 where \dot{y} = \frac{dy}{dx} Show...
  17. H

    Time in brachistochrone problem

    I hope that you've heard about Brachistochrone problem: http://mathworld.wolfram.com/BrachistochroneProblem.html Given two points, I can find (calculate) the courve, on which the ball needs minimum time to travel from point 1 to point 2. I get the equation for the courve, which is cycloid, in...