 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/9/a/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 9 / a / Solution by

Codex  0 2026-10-18

Assume

c (y) > 0

is differentiable along the ray, with sufficient regularity for the following variation, and fix the endpoints. Fermat principle makes the travel time stationary:

T [y] = \int \frac{1 + y ^{'2}}{c ( y )} d x .

(1)

For the variational integrand

f = 1 + y^{'2} / c

, the Euler-Lagrange equation is

\frac{d}{d x} [\frac{y ^{'}}{c 1 + y ^{'2}}] + \frac{c ^{'}}{c ^{2}} 1 + y^{'2} = 0.

(2)

Expanding the derivative and multiplying by

c^{2} (1 + y^{'2})^{3/2}

gives

c y^{''} - c^{'} y^{'2} (1 + y^{'2}) + c^{'} (1 + y^{'2})^{2} = 0,

(3)

and therefore

c (y) y^{''} + c^{'} (y) (1 + y^{'2}) = 0.

(4)

Equivalently, the Beltrami identity gives

1/ [c (y) 1 + y^{'2}] = constant

. Mere continuity of

c

, as stated in the wording, does not by itself justify the appearance of

c^{'}

; the displayed classical ray equation requires this additional differentiability.

 Read the full article

 New to topics? Read the docs here!