= Solution
Vary $r$ by $r+\varepsilon\eta$ with $\eta(a)=\eta(b)=0$. The first <variation> of the <functional> is
$$
\delta I_1=\int_a^b(F_{1,r}\eta+F_{1,r'}\eta')dz=\int_a^b\left(F_{1,r}-\frac d{dz}F_{1,r'}\right)\eta\,dz.
$$
The boundary term from <integration by parts> is zero. The <fundamental lemma of the calculus of variations> therefore gives $\boxed{F_{1,r}-dF_{1,r'}/dz=0}$. For $I_2$, independently vary $r$ and $\theta$ with zero endpoint variations; the two <Euler Lagrange equations> are
$$
\boxed{F_{2,r}-\frac d{dz}F_{2,r'}=0,\qquad F_{2,\theta}-\frac d{dz}F_{2,\theta'}=0.}
$$
For the optical path, <Fermat principle> makes $\int\mu\,ds$ stationary. In <cylindrical coordinates> its <integrand> is $L=\mu(r,\theta)s$, with $s=\sqrt{1+r'^2+r^2\theta'^2}$. There is no explicit $z$ dependence. The <Beltrami identity for several dependent variables> gives
$$
L-r'L_{r'}-\theta'L_{\theta'}=\mu s-\frac{\mu(r'^2+r^2\theta'^2)}s=\frac\mu s=\text{constant}.
$$
The conserved quantity is the axial component of the <optical momentum of a ray>. As $\mu>0$, squaring and inverting yields
$$
\boxed{\frac{1+r'^2+r^2\theta'^2}{\mu^2}=\text{constant}.}
$$
Now suppose $\mu=\mu(r)$ and seek a path of radius $R>0$. The angle is a <cyclic coordinate>, so $L_{\theta'}=\mu R^2\theta'/\sqrt{1+R^2\theta'^2}$ is conserved. At fixed radius this is strictly increasing in $\theta'$, forcing $\theta'=\omega$ constant. The radial <Euler-Lagrange equation> reduces to
$$
\mu'(R)(1+R^2\omega^2)+\mu(R)R\omega^2=0.
$$
Consequently the <helical rays in a radially graded refractive index> have
$$
\boxed{r(z)=R,\qquad\theta(z)=\theta_0+\omega z,\qquad\omega^2=-\frac{\mu'(R)}{R[\mu(R)+R\mu'(R)]}.}
$$
A real nonzero <helix> exists at such a radius when $0<-R\mu'(R)/\mu(R)<1$, and its <helical pitch> is $2\pi/|\omega|$. If $\mu'(R)=0$, the solution is instead a straight axial <light ray>. Thus radial dependence permits constant-radius rays for suitable profiles; an arbitrary radial profile need not support a nonzero real <helix>. This qualification is necessary, for example when the <refractive index> increases with radius.
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