Brachistochrone Equation Problem

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1. Sep 26, 2015

Hamza Abbasi

1. The problem statement, all variables and given/known data
This is the solution of Brachistochrone .

2. Relevant equations

3. The attempt at a solution
I am very confused that how the x in equation(6.24) get its value a(1-cosӨ) ? What is the technique behind this solution of x?

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2. Sep 26, 2015

Staff: Mentor

It is just a clever substitution that replaces x. How to find that:
- in theory: "oh, it is completely obvious that this substitution will lead to a nice solution!"
- in practice: "let's try several approaches until one of them works"

3. Sep 27, 2015

Hamza Abbasi

Why only a(1-cosӨ ? What is the theory behind this substitution?

4. Sep 27, 2015

Staff: Mentor

There is no deeper theory. It is a clever definition of Ө. You can see that it works nicely later.

You could also choose x=Ө2 + 3 cos(Ө) but that wouldn't help.

5. Sep 27, 2015

TSny

To motivate the substitution (6.24), first make the substitution x = au in (6.23). Then complete the square in the denominator. From your experience with integration, you should then see what might be another good substitution to further simplify the integral.