# Paths of minimum time

I am reading An Introduction to the Calculus of Variations by Charles Fox and would be grateful if someone could explain the following bits to me.

1) Legendre test: one of the conditions stated is that (ii) the range of integration $(a,b)$ is sufficiently small. What qualifies as sufficiently small?

2) Section on optical paths: where $\mu$ is the inverse of speed, $ds$ is an element of arc and $\psi$ is the angle the tangent makes with the $x$-axis. Then the characteristic equation becomes $\partial\mu\over\partial y$ = $d (\mu \sin{\psi}) \over ds$ which is independent of the axes. I understand this bit so far, but then comes:

The most convenient system is chosen as follows: Since $\mu$ is a function of $x$ and $y$ only, the equation $\mu=$ constant is that of a plane curve and by varying the constant we get a family of curves which will be called level curves.

I might be interpreting this wrong, but why can we choose $\mu=$constant? And why are we varying it? I have no idea what is happening here. It also says that

If $\mu$ is many valued [through any point there will only be one level curve] if one branch of $\mu$ is adhered to and branch points are avoided.

What "branch"? I have no idea what this statement means.

"sufficiently small" means that if you choose the interval $(a,b)$ small enough, then the statement in question (which you haven't said what it is, but presumably is in the book) will be true. How small it needs to be will depend on other parameters in the statement. – Ted Oct 7 '11 at 8:12