Methods | Part IB, 2004

Starting from the Euler-Lagrange equations, show that the condition for the variation of the integral I(y,y)dx\int I\left(y, y^{\prime}\right) \mathrm{d} x to be stationary is

IyIy= constant I-y^{\prime} \frac{\partial I}{\partial y^{\prime}}=\text { constant }

In a medium with speed of light c(y)c(y) the ray path taken by a light signal between two points satisfies the condition that the time taken is stationary. Consider the region 0<y<0<y<\infty and suppose c(y)=eλyc(y)=e^{\lambda y}. Derive the equation for the light ray path y(x)y(x). Obtain the solution of this equation and show that the light ray between (a,0)(-a, 0) and (a,0)(a, 0) is given by

eλy=cosλxcosλa,e^{\lambda y}=\frac{\cos \lambda x}{\cos \lambda a},

if λa<π2\lambda a<\frac{\pi}{2}.

Sketch the path for λa\lambda a close to π2\frac{\pi}{2} and evaluate the time taken for a light signal between these points.

[The substitution u=keλyu=k e^{\lambda y}, for some constant kk, should prove useful in solving the differential equation.]

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