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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?

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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.

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  • $\begingroup$ Thanks! The fact that Snell's Law follow from Fermat's Principle is known to me for a long time. Strangely, I found many physics student in their third year in university still don't aware to the fact that the two endpoints must be fixed and they even argued with me about it. $\endgroup$
    – ziv
    Commented Mar 13, 2021 at 14:20
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    $\begingroup$ I found additional information about Fermat's principle in a 'Society of exploration geophysicists' wiki. (In the wikipedia article about SEG it is mentioned that most members of SEG are involved in exploration for Petroleum.) In exploration geophysics data from propagation of seismic waves is used. The SEG wiki about Fermat's principle includes description of circumstances where the principle must be applied as a maximum time rule. So it's not inherently least time. More accurate statement: rule of stationary time. $\endgroup$
    – Cleonis
    Commented Mar 13, 2021 at 15:07

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