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.
Please explain. Thanks.