# Calculus of variation

1. Nov 10, 2007

### dakold

Im supposed to show that a ligth beam travelling in a vertical plane satisfies
d^2z/dx^2=1/n(z) dn/dz[1+(dz/dx)^2]. Using calculus of variations to minimize the total time. The vertical plane got a refracting index n=n(z) there z is the vertical position and z=z(x) there x is the horisontal direction.

I have started with to minimize the time and have used Euler-Lagrange equation. I have also simplified and got d^2z/dx^2=1/n(z) {dn/dz+n(z)dz/dx/[1+(dz/dx)^2]}. I don't think this is the same equation as above. is the right way to go or shall i do something else?

thanks