All Questions
Tagged with brachistochrone-problem classical-mechanics
12
questions
3
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0
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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 ...
- 31
1
vote
1
answer
60
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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 ...
- 838
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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 ...
3
votes
1
answer
123
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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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1
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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 ...
- 283
0
votes
0
answers
65
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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 ...
- 56
1
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1
answer
114
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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:
$$ x = \int{\sqrt{\frac{y}{2a-y}} dy} $$
...
- 165
0
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2
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120
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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 ...
- 103
5
votes
1
answer
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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 ...
- 18.8k
3
votes
1
answer
190
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Comparing Brachistochrone curve with a Hypocycloid curve
I want to compare the time that it takes to slide a particle in a frictionless hypocycloid curve, so time would be given by the arclength divided by the velocity
So I need first compute the arclength ...
7
votes
1
answer
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What shape of track minimizes the time a ball takes between start and stop points of equal height?
I was at my son's high school "open house" and the physics teacher did a demo with two curtain rail tracks and two ball bearings. One track was straight and on a slight slope. The beginning and end ...
10
votes
1
answer
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Brachistochrone Problem for Inhomogeneous Potential
This recent question about holes dug through the Earth led me to wonder: if I wanted to dig out a tube from the north pole to the equator and build a water slide in it, which shape would be the ...
- 53.2k