Classical Mechanics (Kibble & Berkshire) · Problem 13.9

Hamilton's equations and phase space — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · fermats-principle

Problem

Consider the propagation of light rays in an axially symmetric medium, where, in a system of cylindrical polar co-ordinates

(r, \theta, z)

, the refractive index

n = n(r)

and the rays lie in the plane

z = 0

. Show that Fermat's time functional has the form

T[r] = c^{-1}\int_{\theta_0}^{\theta_1} n\left(r^2 + \dot{r}^2\right)^{1/2} d\theta,

where

r = r(\theta)

is the equation of the path, and

\dot{r}

means

dr/d\theta

. (i) Show that the extremals of

T

satisfy the ODE

\frac{n\, r^2}{(r^2 + \dot{r}^2)^{1/2}} = \text{constant}.

Show further that, if we write

\dot{r} = r\tan\psi

, where

\psi

is the angle between the tangent to the ray and the local cylindrical surface

r = \text{constant}

, this equation becomes

r\, n \cos\psi = \text{constant},

which is the form of Snell's law for this case. Deduce that circular rays with centre at the origin exist only when the refractive index

n = a/r

, where

a

is a positive constant.