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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Why should we expect the brachistochrone problem to be solvable?
Short version:
Before solving the problem (in particular, before identifying the cycloid as the only possible minimizing curve), is there any physical intuition why we should expect there to be a ...
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Brachistochrone problem with initial velocity
The Brachistochrone problem is usually presented with the having a ball dropped into the slide with initially zero velocity and at position $(x, y)=(0, 0)$.
I would like to know the more general ...
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Which path minimizes travel time for an object moving along a downward parabolic trajectory without friction?
I am studying a physics problem where an object moves from one point to another under frictionless conditions. The trajectory of the object is shaped like a downward parabola, but the exact curvature ...
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Deriving Beltrami identity when there are multiple dependent functions
I'm trying to understand this solution to the brachistochrone problem inside a uniform sphere. Going from equation (19) to (20) and (21), the integrand of the functional we're trying to minimize is of ...
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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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