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Are the endpoints of the light ray path in Fermat's principle must be fixed?

To clarify my question: Using Wikipedia definition for Fermat's Principle:

Fermat's principle states that the path taken by a ray between two given points is the path that can be traversed in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is "stationary" with respect to variations of the path.

Do the variations on the path must not includes any change in the endpoints of the path?

Qmechanic
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Ziv Landau
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1 Answers1

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Yes, the two end points are fixed. If you consider all the different possible paths between the two fixed end points, and allow the speed of light at each point on each path to depend on the refractive index at that point, then the path actually taken by light is the path that minimises the time taken.

In a space where the refractive index (and hence the speed of light) is the same everywhere then the least time path between two points is obviously a straight line. More interesting is the scenario where the refractive index takes one value on one side of a plane, and a different value on the other side. If you apply Fermat’s Principle to paths between two points on opposite sides of the dividing plane you can derive Snell’s Law.

gandalf61
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