Fermat principle and Euler-Lagrange question

In summary, the Fermat principle, also known as the principle of least time, states that light will always take the path that requires the least time to travel between two points. The Euler-Lagrange equation is a mathematical formula used to find the optimal path between two points based on the Fermat principle and takes into account the speed of light and the refractive index of the medium. The Fermat principle is fundamental in optics and explains behaviors such as refraction, reflection, and diffraction. The Euler-Lagrange equation is a crucial tool in theoretical physics for solving problems related to the motion of particles, fields, and waves and has applications in various fields. It is derived using the calculus of variations, which involves finding the path or
  • #1
kkz23691
47
5
Hello,

The Fermat principle says that
(***) Δt = (1/c) ∫ μ(x,y) √1+y'2 dt

Say, we are studying a GRIN material where the refraction index is μ = μ(x,y) and want to figure out the shape of the ray trajectory y=y(x).

Here is what I know (this is not a homework question) but am unsure if the approach is correct:

(1) [itex]\frac{1}{R}[/itex]=[itex]\frac{∂ ln μ}{∂n}[/itex], from Peters' paper . I expect the shape to be a hyperbola say a/x. Can I use this equation (it is derived by seeking stationary time Δt) to calculate μ=μ(x,y)?
(R is the radius of curvature, [itex]\frac{∂}{∂n}[/itex], is a derivative along the normal to the curve)

(2) Once I have μ=μ(x,y), can I plug it into (***), also plug in y=a/x and derive Δt? I am hoping this can then be minimized to find the optimal parameter a or any similar curve parameter.
Does this make sense?

(3) or should I write the Euler-Lagrange equation and try to find out something from it?

(4) or should I use a Lagrange multiplier and "constrain" the ray to the hyperbola a/x

I am unsure which approach is correct; and also, what of the above known/expected expressions should I plug in into the Lagrangian and then do the analysis.
My goal is to find out the extremal curve and its equation.

Need help from someone experienced in Fermat/lagrangian formalism...
 
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  • #2
Thanks in advance!</code>The Fermat principle states that light takes the path of the least time between two points. This means that you should use the Lagrangian formalism to solve for the path of least time.The Lagrangian for this problem is given by L = μ(x,y)√1+y'²dt, where μ is the refractive index.Using the Euler-Lagrange equations, you can solve for the trajectory of the ray. The Euler-Lagrange equations are:d/dx(∂L/∂y') = ∂L/∂yThis gives you the equation of the ray's trajectory.Once you have the equation of the ray's trajectory, you can then plug it into the Fermat principle (***) to calculate the time taken for the ray to travel from one point to another.To optimize the time taken, use a Lagrange multiplier method to constrain the ray to the hyperbola a/x. Here, you will minimize the time taken by varying the parameter a.I hope this helps!
 

What is the Fermat principle?

The Fermat principle, also known as the principle of least time, states that light will always take the path that requires the least time to travel between two points.

What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a mathematical formula used to find the optimal path between two points based on the Fermat principle. It takes into account the speed of light and the refractive index of the medium through which the light is traveling.

How is the Fermat principle related to optics?

The Fermat principle is the fundamental principle of optics. It explains the behavior of light as it travels through different mediums and is essential in understanding phenomena such as refraction, reflection, and diffraction.

What is the significance of the Euler-Lagrange equation in physics?

The Euler-Lagrange equation is a critical tool in the field of theoretical physics. It is used to solve problems related to the motion of particles, fields, and waves, and has applications in various fields such as mechanics, electromagnetism, and quantum mechanics.

How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived using the calculus of variations, a branch of mathematics that deals with finding the path or function that minimizes a specific functional. In the case of optics, the functional is the time taken for light to travel between two points.

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