Minimizing Time with Calculus of Variation in Vertical Plane

[dakold]

Im supposed to show that a ligth beam traveling 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

 

[Jhero]

for your help
Thank you for your question. It seems like you are on the right track with using the Euler-Lagrange equation to minimize the total time. However, there are a few things that need to be clarified in order to fully understand and solve the problem.

Firstly, it is important to specify what the independent and dependent variables are in this problem. From the given information, it seems like the independent variables are x and z, where x represents the horizontal direction and z represents the vertical position. The dependent variable is the refractive index, n, which is a function of z. Therefore, the equation you are trying to solve is a partial differential equation with respect to x and z.

Next, it is important to note that the given equation, d^2z/dx^2=1/n(z) {dn/dz+n(z)dz/dx/[1+(dz/dx)^2]}, is not the same as the one you have simplified to. The correct equation should be d^2z/dx^2=1/n(z) {dn/dz+[n(z)dz/dx]/[1+(dz/dx)^2]}. This is because the derivative of n(z) with respect to z should not be multiplied by dz/dx, as it is a function of z and not x.

Finally, to solve this problem using the calculus of variations, you will need to define a functional that represents the total time taken by the light beam to travel through the vertical plane. This functional will depend on the variables x and z, as well as their derivatives. You can then use the Euler-Lagrange equation to find the minimum of this functional, which will give you the desired equation.

I hope this helps clarify the problem and guide you in the right direction. Good luck with your research!