# Solution to Brachistochrone Problem

• I
• A Physics Enthusiast
In summary, a user is asking for help understanding the Brachistochrone problem and its solution provided by Newton. They have tried searching for resources but are having trouble understanding the solution on Wikipedia. They are looking for a clear explanation of the differential equation involved in the problem and are open to any helpful sources or guidance.

#### A Physics Enthusiast

Can anybody post a full solution of the Brachistochrone problem provided by Newton (with full explanations) ?

A Physics Enthusiast said:
Can anybody post a full solution of the Brachistochrone problem provided by Newton (with full explanations) ?

berkeman said:
Yes. But, I don't think I understand the solution of Newton provided in wikipedia.

A Physics Enthusiast said:
Can anybody post a full solution of the Brachistochrone problem provided by Newton (with full explanations) ?

The short answer to this is no. We can try to help you understand how to solve this problem, but you still have to do the work.

A Physics Enthusiast said:
I don't think I understand the solution of Newton provided in wikipedia.

What are you having trouble understanding?

A Physics Enthusiast said:
Yes. But, I don't think I understand the solution of Newton provided in wikipedia.
If you can describe what you are having trouble understanding then we can probably help.

PeterDonis said:
The short answer to this is no. We can try to help you understand how to solve this problem, but you still have to do the work.
LOL The proof is already there. All I need is a good resource to understand it.

PeterDonis said:
What are you having trouble understanding?
Pretty much everything. I think I understand the Fermat's principle method (till the DE). But, no solution of the DE is provided in wikipedia to conclude that it is indeed a cycloid.

Mind referring a source ?

A Physics Enthusiast said:
The proof is already there. All I need is a good resource to understand it.

The best way to understand it is to solve it yourself. Having a known solution as a guide is often helpful, yes.

A Physics Enthusiast said:
I think I understand the Fermat's principle method (till the DE). But, no solution of the DE is provided in wikipedia to conclude that it is indeed a cycloid.

Which Wikipedia article are you looking at?

This might help:

A Physics Enthusiast said:
I think I understand the Fermat's principle method (till the DE).

Which DE?

A Physics Enthusiast said:
Can anybody post a full solution of the Brachistochrone problem provided by Newton (with full explanations) ?
A Physics Enthusiast said:
LOL The proof is already there. All I need is a good resource to understand it.

I'm confused. If we typed out the proof - which would be the same one in Wikipedia - how would that help you better understand it?

## 1. What is the Brachistochrone Problem?

The Brachistochrone Problem is a mathematical problem in which one has to find the path between two points in the quickest time possible, under the influence of gravity. It was first posed by Johann Bernoulli in 1696 and has since been solved by various mathematicians.

## 2. What is the significance of the Brachistochrone Problem?

The Brachistochrone Problem is significant because it is one of the earliest examples of the calculus of variations, a branch of mathematics that deals with finding the optimal path for a given problem. Its solution also has practical applications in fields such as engineering and physics.

## 3. How is the Brachistochrone Problem solved?

The Brachistochrone Problem is solved using the calculus of variations, specifically the Euler-Lagrange equation. This equation helps to find the path that minimizes the total time taken by a particle to travel between two points under the influence of gravity.

## 4. What was the solution to the Brachistochrone Problem?

The solution to the Brachistochrone Problem was first derived by Johann Bernoulli, who found that the shape of the curve that minimizes the time of descent is a cycloid. This was later confirmed by his brother, Jacob Bernoulli, and other mathematicians.

## 5. What are the practical applications of the solution to the Brachistochrone Problem?

The solution to the Brachistochrone Problem has practical applications in various fields, such as designing roller coasters and finding the optimal path for objects to travel in space. It also serves as a basis for other problems in the calculus of variations and has helped advance our understanding of the mathematical concepts involved.