We know Newton's law doesn't hold in very small distances and very high speeds. Do you know any conditions under which the principle of least action doesn't hold/breaks down?
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Possible duplicates: https://physics.stackexchange.com/q/3500/2451 , https://physics.stackexchange.com/q/20298/2451 and links therein. – Qmechanic Mar 09 '22 at 06:14
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In quantum systems, the classical action is not minimized (which is not particularly surprising, given that it is classical). By means of the path integral formulation of Quantum Mechanics, one learns that to compute the transition probabilities of a system we have to consider all of the possible "histories" of the system, similar to what happens in the double slit experiment. However, it can be shown that for "large systems" the greatest contribution to the sum over histories comes from the history with extremized classical action, and hence one recovers the classical Principle of Least Action. This is discussed in the Feynman Lectures on Physics, particularly on Sec. 26.6 of Vol. I and Chap. 19 of Vol. II.
It should be pointed out, however, that one can restore the Principle of Least Action in quantum mechanics by considering instead the effective action (also known as quantum action), which already includes quantum effects in its expression. It is, however, impossibly harder to compute exactly.
As for Relativity, there are no problems with the Principle of Least Action. In fact, even General Relativity comes from an action.
Níckolas Alves
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