Brachistochrone Problem: The Cycloid Solution

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SUMMARY

The Brachistochrone problem determines the fastest path between two points under the influence of gravity, which is a cycloid. This conclusion arises from mathematical derivations that show the cycloid's unique properties, allowing for maximum acceleration at the start of the descent. The path's steep initial angle enables an object to gain significant velocity, maintaining it throughout the trajectory. The discussion emphasizes that while other curves may share similar characteristics, the cycloid is mathematically proven to be the optimal solution.

PREREQUISITES
  • Understanding of calculus, particularly optimization techniques.
  • Familiarity with parametric equations and their applications.
  • Basic knowledge of the principles of physics, specifically gravitational acceleration.
  • Concept of cycloids and their geometric properties.
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  • Study the derivation of the Brachistochrone problem using calculus of variations.
  • Explore the properties of cycloids and their applications in physics and engineering.
  • Learn about the relationship between acceleration and path shape in classical mechanics.
  • Investigate other optimal path problems in physics, such as the Tautochrone problem.
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Mathematicians, physicists, engineering students, and anyone interested in classical mechanics and optimization problems.

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Brachistochrone derivation

About the brachistochone---

I have calculated the parametric equation for it...but why is this path a cycloid? It does seem fast...but how did they determine that it was a cycloid...not any other function?--Hmm-why exactly does the path of a point of a circle as it rolls down a straight line become the fastest distance between the two points in the brachistrone problem??
 
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Why is it a cycloid? Because it makes 'the best use' of the gravitational force. At first the path is very steep, so the object acquires a large acceleration, and because teher is no friction involved the object keeps this large velocity till the end of the track. Ofcourse there are other paths with the same characteristics, and I think it is a cycloid because this follows from the formulas.

If there is a connection between the shape being the one of a piont on a horizontally moving wheel, and the brachistochrone I would definitely want to hear more...!
 
bomba923 said:
About the brachistochone---

I have calculated the parametric equation for it...but why is this path a cycloid? It does seem fast...but how did they determine that it was a cycloid...not any other function?--

Mathematics and only mathematics.It could have been any other curve similar in shape,but it was the equations that led to the solution.

bomba923 said:
Hmm-why exactly does the path of a point of a circle as it rolls down a straight line become the fastest distance between the two points in the brachistrone problem??

There's no connection between the cycloid followed from a point on rolling wheel and the brahistochrone.The point would describe the same cycliod even in the absence of gravity.

For more detail,check out this wonderful site:you have a model of brahistochrone right on the first page
site
It's in french...But it's math.It's comprehendable.

Daniel.
 
da_willem said:
If there is a connection between the shape being the one of a piont on a horizontally moving wheel, and the brachistochrone I would definitely want to hear more...!

That's was my question in the first place--! Anyways, I'll try to figure it out
 
Please,do tell us if u find it.It would be a really interesting both mathematically and physically result.


Daniel.
 

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