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It is on the basis of Principle of Least Action, that Lagrangian mechanics is built upon, and is responsible for light travelling in a straight line.

Is its the classical equivalent of Schrodinger's equation?

Qmechanic
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megamonster68
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As to the title question: no, the principle of least action is not a first principle. A first principle is a kind of fundamental principle rooted directly in empirical science. For example, no one has measured in some experimental scheme the general validity of the principle of least action, rather it is a principle that merely known to work in a vast number of special cases. The constancy of the speed of light or the fact that light travels in straight lines, however, have been generally verified, and are established principles of empirical physics. Similarly, the Schrodinger equation has been verified as fundamental to quantum theory and is thus a first principle of quantum mechanics. The principle of least action is certainly fundamental to the methodologies of theoretical physics, e.g. Schrodinger originally used variational methods to derive his famous equation, however, it is not an empirical first principle in the strictest sense.

Albertus Magnus
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  • The idea of a first principle is something from which a theory can be derived. It does not necessarily have to be something that can be directly tested experimentally. The Schroedinger equation is derived from Planck's relationship and therefore does not qualify as a first principle. – flippiefanus Feb 16 '24 at 03:55
  • @flippiefanus I understand first principle as a philosophical concept, however, scientists are usually in the habit of rooting first principles in empiricism because they are unable to get around an empirical fact, it is a truncating point in the logical series. A person can continue the process of hunting for first principles in the manner as yourself ad absurdum; this was the whole point established by Pyrrhonian skeptics. Scientist are providing an alternative to sophistry. – Albertus Magnus Feb 16 '24 at 12:53
  • The philosophy of science is intrinsically inclined to the useful, regardless of how much illustrious thinkers like Dirac go on about beauty. – Albertus Magnus Feb 16 '24 at 12:59
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In a sense, the principle of least action can be derived as the classical limit of the path integral formulation of quantum mechanics, which by extension becomes its quantum equivalent. In this view, the principle is not a first principle, but a consequence of destructive interference between different propagation paths.

In other words, according to quantum mechanics, the light way takes "every possible path at the same time", but the non-stationary paths interfere destructively, causing their amplitudes to disappear, leaving only the stationary path (the "least action" path). This interference can be observed empirically through e.g. the double slit experiment.

Codename 47
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  • You got it the wrong way around. Feynman developed the path integral formulation based on Fermat's principle, which is an optical version of the principle of least action. – flippiefanus Feb 16 '24 at 03:57
  • I didn't mean to imply any specific historical order to the results, just that since quantum mechanics is more fundamental than classical mechanics, the principle of least action can be seen as a limit of the path integral formulation, even if they were developed in reverse order. – Codename 47 Feb 16 '24 at 09:45
  • The principle of least action is not restricted to classical mechanics. That is why it works for the path integral in the quantum context. The latter is a formalism, not physics per se. – flippiefanus Feb 17 '24 at 03:50
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You say, "Is its the classical equivalent of Schrodinger's equation?" so you seem to be wanting to work in an (imaginary) classical world. That's fine. The Schrödinger equation isn't fully general (it's nonrelativistic) either.

However, even under these restricted conditions, there is no unique way to choose what is a "first principle". As in mathematics, you often have a choice as to what statements you choose as your axioms and which are derived.

You can assume the Principle of Stationary Action (it's not always a minimum) and derive the Hamiltonian and Hamilton's equations from that. Or you can start from Hamilton's equations as your first principles and derive the Lagrangian formalism and the Principle of Stationary Action. As far as I know, they are equivalent and neither is more general than the other. Which is more parsimonious is a matter of taste.

In fact, you can do the same thing with the Schrödinger equation. You can instead start from matrix mechanics and derive the Schrödinger equation.

WaterMolecule
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    I concur with the importance of specifying that since Hamilton's action is not always a minimum a better name for it is 'Stationary action. About the 2014 question that you linked to: 'When is the principle of stationary action not the principle of least action?'. I added an answer there, because the existing most voted answer offers just a single example, rather than a general rule. In the answer I added I give the rule that can be used to see in advance whether a minimum or maximum will be encountered – Cleonis Feb 17 '24 at 18:24