How Does x in Equation 6.24 Become a(1-cosӨ)?

[Hamza Abbasi]

Homework Statement

This is the solution of Brachistochrone .

Homework Equations

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?

 

[mfb]

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"

 

[Hamza Abbasi]

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

 

[mfb]

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.

 

[TSny]

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

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.