Brachistochrone problem Definition and 18 Threads

Conservation of energy is invoked, which I understand, but the author omits most of the steps. He arrives at (18): I've tried to substitute (13) and (17) into the LHS of the energy conservation equation, but the result is quite messy and my gut feeling is that it doesn't simplify in a nice...

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...

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...

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...

Brachistochrone problem’s original proof by Snell’s law and a story of Newton.

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 !

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...

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...

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

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...

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?

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...

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...

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 =...

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...

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...

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...

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...