After Einstein’s creation of the general theory of relativity, it became clear that time in different points of space can flow at different speeds because of the influence of gravity. Say, at point A, the hydrogen atom emits light with one frequency, at point B it emits with the other, in C does with the third. Therefore, standard (identical) atomic clocks at different points of the gravitational field can go at different rates. New concepts arose: local time, world (laboratory) time, proper time. If Fermat now formulated his principle, he would be asked:
The light moves to spend a minimum of time, by what clocks?
In addition, after creation of quantum mechanics, the physical meaning of the principle of least time became clear. The motion of a particle in quantum mechanics is described by a wave function, which can be represented as a sum of plane waves. The wave propagation path is determined by the following condition: the phase difference at the end and beginning of the path must be minimal. Here is what is written about this in the Course of Theoretical Physics of Landau and Livshits, Quantum Mechanics, Chapter 1.6. The passage to the limiting case of classical mechanics:
So, the principle of least action is connected with Fermat’s principle. However, when we tried to apply Fermat’s principle to the motion of a solid body, we got the wrong result. This means that we have not understood this subject well. We need to better understand both these principles. For this it is worthwhile to use something very simple and very fundamental.
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