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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.

Qmechanic
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2 Answers2

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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.

Miyase
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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.

Mattia
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