3. Fermat's principle says that a light ray travels along the path requiring the least time to join endpoints. Further, the speed of light changes with the density of the medium, slowing as the medium becomes more dense. These observations are combined into the following: Definition. The optical length of a path a(s) taken by a ray of length from a (0) to a(() is given by F(a) = n(a(s)) ds, where n(x) is the index of refraction of the medium at the point x in space. and Theorem. Light follows the path of least optical length between a (0) and a((). (1) Suppose that we have a parametrization o (t) which is not by arclength. Use u-substitution to change variables from s to t in the optical length integral."\f(2] Now suppose that D:I[x) = (x, y{x]) and n(x, y) = %. Over the next few parts, we will derive a formula for the path of the light ray from {0, I) to (I, I]. Write the optical length functional as an integral in the form 1 Fa) = f forw- and then nd the Hamiltonian .' I I a lm J = foxy ) -}2 afya Simplify your answer as much as possible. You may assume that y 2:: D, but you can't make any similar assumption about y\". \f(4) Now find the general solution of the first order ODE you derived in the last part for y(x). Your solution will involve another unknown constant D. You may continue to assume that y >0."This is an autonomous ODE, so we have a well-practiced method for solving the equation.(5) Your solution to the Euler-Lagrange equation should involve two unknown constants. Use the boundary conditions y(0) = y(1) = 1 to solve for the constants and draw a sketch of the path of the light ray.(6) Suppose that H. Potter is located at (0.5, 1) while A. Filch is located at (0, 1) looking at a tapestry located at (1, 1). Will Filch see Potter? Why or why not?"Physicists use this idea to build "optical cloaking" devices. See https://www.youtube. com/watch?v= oJb9RnAVDuE for a really great demonstration