Hacker News Reader

> It's extremely unintuitive that classical systems should minimize anything like a time or an action.

If I get to define the measure arbitrarily, then I can always find a measure that something else is always a minimum of. So in that sense it's not surprising at all. The interesting question to me is why should that seemingly arbitrary measure be action? What does that even mean, physically? I have no intuition for it.

> Think about it: they travel along the minimum path,

I'm already stuck here. What would it even mean for a particle to have an "action" (whatever that is) that is not minimized? Like what would that look like, physically? I understand what it means for distance not to be minimized, but action isn't distance...

>What would it even mean for a particle to have an "action" (whatever that is) that is not minimized?

The particle doesn't have an action. The trajectory of a particle is what the action is defined in terms of. One way to think of it would be "it's a measure of how much the trajectory deviates from the one dictated by Newton's equations." Pretty much like what you said: "I can always find a measure that something else is always a minimum of."

About what a trajectory with non-minimal action would look like: it would be an arbitrary violation of the equations of motion for the system (ex: free particle moving in a zigzag instead of a straight line at constant velocity). Moving in a straight line at constant velocity is what Newtonian mechanics prescribes, and that trajectory will minimize action for the corresponding Hamiltonian.

> then you kind of already accept that (ordinary, classical) light somehow knows the time it takes to go along all possible paths, then chooses the minimum

But it doesn't, minimum (or more precisely extremum) is a local quality. Basically light goes by the path with zero derivative because otherwise neighboring pathes interfere. Feynman lectures touch on it relatively early [1] which I think is nice

[1] https://www.feynmanlectures.caltech.edu/I_26.html#Ch26-S5

It is not local. That’s what’s wild about it. It’s a minimum over all paths. Re-read that section you posted.

> In other words, the principle is that light takes a path such that there are many other paths nearby which take almost exactly the same time.

> the wavelength tells us approximately how far away the light must “smell” the path in order to check it.

For the same reason, it doesn't even have to minimum at all, it should be extremum.

Many other responses are very good.

> It's extremely unintuitive that classical systems should minimize anything like a time or an action.

Perhaps, they minimize the action as the primary driver (cause), and time (effect) is generated as part of the solution, as a definition of evolution ...

This margin is too small to contain my full explanation :)

I have written above a more complete explanation.

Hamilton did not discover any "action". Like everybody since Maupertuis, a century earlier, Hamilton used "action" with the meaning "accumulated living force".

"Living force" is English for the Latin "vis viva", which is the old name of kinetic energy (the term "kinetic energy" was introduced only later, in 1854, by William Thomson, who in 1892 became Lord Kelvin; for a short time before 1854 "actual energy" was used instead of "kinetic energy" and opposed to "potential energy", as defined by Rankine in 1853). So "action", in the sense introduced by Maupertuis and used by everybody until the 20th century, meant integral of the kinetic energy (actually the living force was mv^2, so the double of the kinetic energy).

Hamilton has introduced a new physical quantity, never used by anyone before him, which he named just "function S". Unlike the principle of the minimum action, which is false except for certain restricted cases, Hamilton's variational principle about the "function S" is always true, including in relativistic mechanics and quantum mechanics.

Nowadays Hamilton's "function S" is usually called "Hamilton's action", because it has the same measurement unit like the traditional action, even if it is a different physical quantity. "Hamilton's" is frequently dropped, which does not cause much ambiguity, because now the traditional "action" is seldom mentioned.

Nevertheless, whenever history is discussed, a very clear distinction between "action" and Hamilton's "function S" must be maintained, otherwise it is impossible to understand the evolution of physics.

Hamilton has discovered his function S by starting from the system of equations of Lagrange and finding a way to deduce them from a simpler principle.

It is a little weird that even if Lagrange had discovered when young, together with Euler, the Euler-Lagrange equations for variational problems, many years later, when he has written his works about mechanics, he has never attempted to use any variational techniques in the formulation of his equations of dynamics (and he dismissed the principle of the minimum action as seldom applicable), so this relationship has been discovered only later, by Hamilton. While Lagrange has been the first who has used correct definitions for the kinetic energy and the potential energy, he has named them just "fonction T" and "fonction V", similarly to Hamilton's use of just "function S".

> how do they know that that path is the minimum? Do they sample the other paths to know?

Isn’t it just a tautology? After all, how do we measure shortness and straightness?

Least action is a nice, unifying principle, but to me it strongly hints that our formalisms don't capture the full picture.