Optics in Lobachevsky geometry

[raopeng]

Homework Statement

In the upper half of a (x,y) plane endowed with a refractive index of n(y) = 1/y, find the form of light ray.

Homework Equations

l = ∫n dl

The Attempt at a Solution

My method is to construct a functional for optical path, obtaining the result using Euler-Lagrangian equation which would give the desired result of circle with arbitrary radius(the geodesic of the Poincare model). However as I find this question in Arnold's ODE, is it possible to start from a more "ODE" prospective, because I believe Arnold didn't expect the use of Hamilton's principle from this question...

 

[mighty2000]

Thank you for your interesting question. I would approach this problem by first understanding the concept of refractive index and how it affects the path of light. Refractive index is a measure of how much a material bends light as it passes through it. In this case, the refractive index is dependent on the y-coordinate, which means that the bending of light will vary as it moves along the y-axis.

To find the form of the light ray, we can use Snell's law, which states that the angle of incidence is related to the angle of refraction by the refractive indices of the two mediums. In this case, we have a varying refractive index, so we can use the concept of a continuously varying medium to solve this problem.

Using the equation l = ∫n dl, we can calculate the optical path of the light ray as it moves along the y-axis. This integral takes into account the varying refractive index, which will affect the bending of the light ray. By solving this integral, we can obtain the desired form of the light ray, which will be a circle with an arbitrary radius, as you correctly stated.

In conclusion, I believe that understanding the concept of refractive index and applying it to the problem at hand is the most appropriate approach to solving this question. I hope this helps and feel free to ask any further questions. Good luck with your studies!