Questions tagged [brachistochrone-problem]
the problem of finding the path between two points such that the transit time under specified conditions is minimized.
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Find curve minimizing energy loss due to friction [closed]
I am looking for an ansatz of the following problem:
Given a mass $m$ moving in a constant gravitational field along curves $C$ connecting two fixed points I want to find the curve $C_0$ that ...
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Relating Brachistochrone problem to Fermat's principle of least time [closed]
When I came across the Brachistochrone problem, my teacher said we could relate it to Fermat's principle of least time.
So, we could make many glass slabs of high $\mathrm dx$, and every slab has a ...
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How to prove that the Brachistochrone problem could be reduced to finding a curve on a plane?
Given two points in space, the 2D Brachistochrone problem could be solved to give solution of a cycloid. I am wondering how could one prove that in arbitrary dimensions ($d\geq 3$) with a 1D uniform ...
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Which block reaches the floor first?
There are two blocks, each starting at the top of an incline. The particular inclines are depicted in the image below.
The height through which the blocks fall is the same, the table lengths are the ...
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Inconsistency in solving the Brachistochrone Problem. Did I make a mistake? [closed]
Background: Equation of Motion
Okay. First I want to see if my "Newtonian Mechanics" lens of the problem is correct.
Let the particle's path be given by $\vec{r}(t) = (x(t), y(t))$ and just ...
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Variation of functional with respect to Lagrange multiplier in QM
So, I am reading a paper on Quantum Brachistochrome and on the second page they say that they are doing a variation w.r.p. $<\phi|$, (which is a lagrange mulriplier) of the following action:
$$ S(\...
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Fermat least time and snell's law for multiple layers of medium
I am reading a differential equation book that discusses the Brachistochrone problem. The book discusses Bernoulli's solution that uses Snell's law. The book says that a ray would follow the fastest ...
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Solution of Brachistochrone Problem with friction
$\def \b {\mathbf}$
solution of Brachistochrone Problem with friction
from
https://mathworld.wolfram.com/BrachistochroneProblem.html
I found the EL equation (29) and the parametric solution equations $...
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Non-differentiable solution of the Brachistochrone problem
Is there a solution to the brachistochrone problem where the solution is non-differentiable everywhere (angular point)?
The Euler-Lagrange method fails if the first or second derivative of the ...
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Why do I need the Beltrami identity to solve the brachistochrone problem?
Brachistochrone problem
The time to travel from point $p_1$ to $p_2$ is given by this integral
$$t_{12}=\int_{p_1}^{p_2}\frac{ds}{v}.$$
With $ds=\sqrt{dx^2+dy^2}=\sqrt{1+y'^2}\,dx$ and $v=\sqrt{2g\,y}$...
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What is the definition of a Brachistochrone curve in a non-Euclidean space?
I have a problem where I have to study "the geometric properties of the Brachistochrone curve in non-Euclidean spaces". But I am confused about the definition of the Brachistochrone Problem/...
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Can a frictionless brachistochrone provide maximum range when projecting a mass on exit?
A puck is released from the top of curved, frictionless track. The puck descends, then rises again at the end, such that it leaves the track and continues in free fall until hitting the ground. The ...
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Evaluating the integral in the brachistochrone problem numerically
When solving the brachistochrone problem (path of least time for a mass sliding on the path, with the path having no friction, from point A to point B), the solution curves are solved from the ...
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Brachistochrone problem with drag
The original Brachistochrone problem is without friction and drag. The Brachistochrone problem can also be solved analytically with friction. But what would the optimal path be if there was a drag ...
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Exact function for the brachistochrone
I have watched videos on the brachistochrone problem and how to find the quickest path a particle can take between two points. However they never gave an exact function for the path. I thought of ...
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For virtual displacement in the Lagrangian, why is $\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0$?
I am having trouble understanding why $$\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0.\tag{7.132}$$
you can see my explanation leading up to it below.
I would greatly ...
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Is this idea/concept very useful to alternatively tackle a brachistochrone problem?
As we know initially we would be given two points on a 2D surface (consider a plane perpendicular to ground ) we would like to run a small ball from higher level point to lower level point from a ...
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Hypocyloid Integral in Polar Coordinates
I've been working on the classic problem of finding the path through which a body travels in least time between two points on the surface of the Earth, assuming that the body is allowed to fall ...
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Brachistochrone Problem without Trigonometric Substitution
I'm trying to numerically reproduce the cycloid solution for the brachistochrone problem. In doing so, I eventually ended up with the following integral:
$$ x = \int{\sqrt{\frac{y}{2a-y}} dy} $$
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Brachistochrone to a vertical line [closed]
Just for fun, I am working through some problems in Mathematics of Classical and Quantum Physics by Byron and Fuller. Problem 2.13 reads:
Prove that a particle moving under gravity in a plane from a ...
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Explanation for extra portion of Brachistochrone curve?
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 ...
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Another Solution To Brachistochrone Problem
Recalling the statement of the problem :
Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the ...
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Descent on an inclined wavy frictionless track [closed]
The classical Brachistochrone was actually counterintuitive wherein the time of descent is lesser (the least) for the cycloid than that of the corresponding straight inclined track.
Let an inclined ...
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Could two concatenated cycloids be an optimal solution to the Brachistochrone problem?
The following is a specific instance of the brachistochrone
problem, which I first encountered in grad school, and I
have occasionally used as hw problem in teaching CM.
A particle is started from ...
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What is the direction of the velocity $v=\sqrt{2gy}$ vector? Is it tangential? If so, why?
I was trying to obtain the brachistochrone as a function of time, and I failed several times because I wrongly assumed that the $v=\sqrt{2gy}$ vector points downwards (vertical vector). However, in ...
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What is the intuitive reason that the trajectory of a charged particle in a uniform crossed electro-magnetic field is a brachistochrone?
Cycloid is a type of trajectory which is traced by a point on the circumference of a planar circle rolling without slipping on a surface. It turns out that this is the solution to the Brachistochrone ...
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Is a brachistochrone a straight line in curved space?
Please bear with me, and don't get upset if i have lack in knowledge about spacetime.
Brachistochrone:
Given two points A and B in a vertical plane, what is the curve traced out by a point acted ...
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Brachistochrone Problem for Spacecraft
I understand how the Brachistochrone Problem works, and do understand how the friction is added to it, however I am unsure of how to use the Brachistochrone Problem for use in spacecraft.
We know ...
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Acceleration downhill, fastest trajectory for a ball
Given 3 ways of going downhill, like in this image:
Would a ball behave like that in real life? Intuitively, it makes no sense. The shortest path here is not the fastest.
Any hints to the math ...
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Curve for fastest time down a ramp [duplicate]
I came across a physics experiment video showing three balls released from a point A, going down three different kinds of ramps leading to a point B (https://www.youtube.com/watch?v=61S0KW7e-rc)
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