Brachistochrone problem with friction

In summary, the conversation was about the brachistochrone problem with frictions and a demonstration found on the web. The person asked for help understanding the steps between equations (29) and (30) and (30) and (32), (33). Another person explained that to go from (29) to (30), one needs to integrate and to convince themselves that (30) is the result of integrating (29). The original person thanked the other for their help. Another person later joined the conversation, asking for help with steps (28) to (29) and how it relates to the Euler Lagrange equation.
  • #1
janonus
3
0
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 nice !
thanks :)

The link to the article : http://mathworld.wolfram.com/BrachistochroneProblem.html
 
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  • #2
To go from 29 to 30, it is pretty clear that they have integrated one time. Eq(29) is second order while eq(30) is first order. What you need to do is convince yourself that 30 is indeed the result of integrating 29.
 
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Likes janonus
  • #3
Yeah, I got it !
Tanks for your help Dr.D ! :smile:
 
  • #4
Hi, not sure if it is ok to revive an old question, I am also having trouble following some of the steps in this demonstration, in particular steps (28) to (29). I know what the Euler Lagrange equation is, but applying it to the integrand in (28) does not appear to simplify to (29). Would anyone be able to help me with this?
 

Related to Brachistochrone problem with friction

1. What is the Brachistochrone problem with friction?

The Brachistochrone problem with friction is a mathematical problem that involves finding the path of quickest descent between two points in the presence of friction. It is a variation of the Brachistochrone problem, which only considers the effects of gravity.

2. Who first proposed this problem?

The Brachistochrone problem with friction was first proposed by Johann Bernoulli in 1696. He challenged his brother, Jakob Bernoulli, to find the curve that a bead would follow from one point to another in the shortest amount of time, taking into account the effects of friction.

3. What is the solution to this problem?

The solution to the Brachistochrone problem with friction is a cycloid curve, which is the same as the solution to the Brachistochrone problem without friction. However, the addition of friction makes the path slightly longer and slower.

4. How is this problem relevant in real life?

The Brachistochrone problem with friction has practical applications in various fields, such as engineering, physics, and transportation. It can be used to optimize the design of roller coasters and other amusement park rides, as well as the trajectories of objects in motion, such as projectiles and vehicles.

5. Are there any generalizations of this problem?

Yes, there are several generalizations of the Brachistochrone problem with friction, including the Tautochrone problem (finding the curve that a bead would follow in the same amount of time, regardless of its starting position), the Isoperimetric problem (finding the shortest curve between two points that encloses a given area), and the Brachistochrone problem with multiple frictional forces (considering the effects of multiple sources of friction).

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