How does Fermat's Principle follow from Huygen's principle?

In summary, there is a proven connection between Fermat's principle of least time and Huygen's wave principle. This connection is discussed in Chapter 10 of the Notes on Analytical Mechanics, where it is shown that these two principles are equivalent. However, despite being commonly mentioned, there is a lack of available proof of this connection.
  • #1
nhmllr
185
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How does Fermat's principle of least time (that if light goes from one point to another, any small deviation in the path will result in a path that takes the same time on the first order) follow from Huygen's wave principle?

Everyone always says that Fermat's principle can be derived from Huygen's principle for when the distances you're looking at are much bigger than the wavelength, but no one actually every proves it.

I'm giving a little talk to my class about Fermat's principle, and I'd like to connect it to Huygen's principle. Can anyone direct me to a treatment, because I can't find one.

Thank you
 
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  • #2
This is done for you in Chapter 10 - Connections to Optics of my Notes on Analytical Mechanics. Fermat's Principle, the eikonal equation, and Huyghens' wavelets are shown to be equivalent.
 

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1. What is Fermat's Principle?

Fermat's Principle is a principle in optics that states that light will always take the path of least time when traveling from one point to another.

2. What is Huygen's Principle?

Huygen's Principle is a principle in optics that states that every point on a wavefront can be treated as a source of secondary waves, which combine to form the next wavefront.

3. How does Huygen's Principle relate to Fermat's Principle?

Huygen's Principle is the basis for the derivation of Fermat's Principle. It provides a way to visualize and understand why light takes the path of least time.

4. Can you explain the mathematical relationship between Huygen's Principle and Fermat's Principle?

Huygen's Principle can be mathematically derived from Fermat's Principle using the principle of least action. This means that the path that light takes must minimize the action, which is defined as the integral of the Lagrangian over time.

5. How does understanding the relationship between Huygen's Principle and Fermat's Principle help in understanding the behavior of light?

Understanding this relationship allows us to predict and explain the behavior of light, including phenomena such as refraction, reflection, and diffraction. It also helps in developing new technologies such as lenses, mirrors, and optical fibers.

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