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I have gone through some of the questions asked here re Hamilton's principle, but could not readily find an answer to the following:

Hamilton's principle states that paths particles follow extremizes the action.

I'm not confused (I think) about how to derive the Euler-Lagrange equations. Rather, I'm confused how nature 'knows' beforehand (for instance at $t_i$) what the extremal path is between $q(t_i)$ and $q(t_f)$?

Note that my question is in a classical setting, perhaps a quantum mechanical setting will clarify it - I don't know.

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    $\begingroup$ this may help you $\endgroup$
    – Brian
    Commented Jun 11, 2022 at 10:50
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    $\begingroup$ Why does "nature" need to "know" anything, and what do "nature" and "know" in this question even mean? $\endgroup$
    – ACuriousMind ♦
    Commented Jun 11, 2022 at 11:01
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    $\begingroup$ @ACuriousMind Well, I suppose that there are people that do ponder why a priori the particle will take the extremal path when at $t_i$ it is confronted with an infinite number of paths. $\endgroup$
    – user228424
    Commented Jun 11, 2022 at 11:09
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    $\begingroup$ Nature is Hamitnton's principle. $\endgroup$
    – Lewis Miller
    Commented Jun 11, 2022 at 11:16
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    $\begingroup$ Physics doesn't answer "why" questions. $\endgroup$
    – Lewis Miller
    Commented Jun 11, 2022 at 11:19

2 Answers 2

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This is a rather deep question. In classical physics, you have to be pragmatic: the theory gives results that line up with reality, so we're happy.

From a quantum point of view, I'd suggest you have to look at the path integral formalism. Roughly speaking, probability waves explore all paths, interfere with each others, and only the "real" path remains.

In this formalism, nature doesn't have to know anything in advance, the selection happens "automatically" through interference.

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    $\begingroup$ Yes, somehow a statement along the lines of on average the path taken is extremal would make more sense to me. I have yet to study (actually refresh my little knowledge of) path-integrals, but I suppose the path-integral formalism works with probabilities. $\endgroup$
    – user228424
    Commented Jun 11, 2022 at 11:07
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    $\begingroup$ I tried to keep my answer concise. If you go through the details, you'll see that it's a bit more complicated than that, in part because those waves don't really "live" in normal 3D space. $\endgroup$
    – Miyase
    Commented Jun 11, 2022 at 11:08
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It might be a circular reasoning, but, if the path that extremizes the action satisfies the Euler-Lagrange equations, there is at every time a local condition that the path must satisfy without looking at the equivalent global condition. In some cases, like collisions in which the path is not smooth (so differential equations are senseless), only the Hamilton principle holds. Hamilton principle is useful to extend theory beyond the validity of the Euler-Lagrange equations, so in these cases my answer couldn't be valid. But in the case of collisions the collision time is arbitrarily small, so again local.

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  • $\begingroup$ Thank you - I had the same kind of reasoning which I also found a bit circular, namely that if E-L are satisfied then the path will be extremal. $\endgroup$
    – user228424
    Commented Jun 11, 2022 at 11:31
  • $\begingroup$ @FridoRolloos Yeah it could be unsatisfactory but the reasoning is valid. It's senseless asking if the nature 'know' the E-L-E or the Hamilton principle in the cases in which they are equivalent: if two theories provide the same results there is no way to distinguish between them. $\endgroup$
    – Mattia
    Commented Jun 11, 2022 at 23:29