In the Principle of Least Action, how does a particle know where it will be in the future?

Question

In his book on Classical Mechanics, Prof. Feynman asserts that it just does. But if this is really what happens (& if the Principle of Least Action is more fundamental than Newton's Laws), then don't we run into some severe problems regarding causality? In Newtonian Mechanics, a particle's position right now is a result of all the forces that acted on it in the past. It's entirely deterministic in the sense that given position & velocity right now, I can predict the future using Newton's laws. But the principle of least action seems to reframe the question by saying that if the particle ends up in some arbitrary position, then it would take a certain path (namely one minimises the action). But that means that the particle already knows where it'll be and it "naturally" takes the path that minimises the action.

Is there any deeper reason for why this is true? In fact principle of least action seems so arbitrary that it's hard to see why it manages to replicate Newton's Equations? If any of you have any insight into this, please share because I just cannot get my head around it.

Note - Please keep in mind, my question is regarding the principle itself, not the equations that result from that principle.

Answer 1

The particle doesn't have to "know" anything. The principle of least action is used when we already know the endpoints of the path, and we want to find out how the particle got from the initial to the final position. We need to specify the final position in advance.

This makes it look like least action can't make predictions about the future. However, that's not a problem in practice, because given an action principle with local action, you can derive a local differential equation, called the Euler-Lagrange equation, that holds at every point for every legal path. In most practical calculations, we work with the Euler-Lagrange equations directly.

So you can think of these action principles locally or globally, and the results will be exactly the same, even if the philosophical window dressing is different. I think Feynman really liked the picture of a "smart" particle "sniffing out" which paths to take, so he tends to explain it globally.

Answer 2

Imagine you have an initial position.

Then there are lots of different possible initial velocities and those different velocities might have you end up in different places. So a different initial velocity might give you a different final position.

So instead of describing the different initial velocities you could instead describe the different final positions.

That's what the principle of extremal action does. Instead of fixing the initial velocity it fixes the final position. In the end you get an equation of motion, and you never even had to say what the final position was. And so as long as there was a final position (i.e. the particle doesn't cease to exist before $t=t_f$) then everything works out fine.

And not every potential plus initial position plus initial velocity gives a unique solution according to Newton's Laws. You could try to propose that as Newton's zeroeth law but firstly, that's historically misleading, and even so what do you restrict, certain initial positions, certain initial velocities, certain potentials?

But neither the principle of extremal action nor Newton's laws of motion require that there be a unique solution, they just predict the actual solution satisfies an equation.

Answer 3

Do not take Feynman's metaphorical language at face value. There are neither classical "particles" nor classical "causality" in quantum theory, which presumably describes what "really" happens, both are artifacts of the classical description. And in classical description the only physically relevant fact is that classical trajectories have to obey classical laws, equivalently expressed in Newton's or least action form. The rest like "deterministic" or "particle knows" are just literary devices used to explain what those laws state. We can infer particle's possible future states from the current one from Newton's laws, but we can also solve Newton's equations in reversed time, and infer its possible past states instead. This does not "reframe" Newton's laws from causality to teleology or vice versa, nor are they "reframed" when we re-express them in the form of an extremal principle, see Disputes about possible teleological aspects.

And "the principle of least action" is a misnomer, the trajectory must be an extremal of least action, not necessarily minimize it (the difference is the same as between critical points and maxima/minima for functions). And the former unlike the latter is a local condition, i.e. the particle need not "know" its destination to extremize the action, which is why the trajectories can be computed locally from Newton's laws.

Answer 4

Imagine rainwater falling on a crinkly mountain. Each raindrop follows deterministic Newtonian mechanics, and follows the 'easiest' path downhill, with successive drops forming 'trajectories' and flowing into streams.

This is basically the Principle of Least Action. Each drop obeys local forces, but over a long interval of time - the time integral of Action is minimal, when compared with nearby alternative paths.

Answer 5

I'm fairly sure that Feynman's point has nothing to do with comparing Newton's Laws to Lagrange's equation or to the Hamiltonian equations of motion in the context of classical mechanics and everything to do with the ability of Lagrangian and Hamiltonian mechanics to be extended to cover field theories like those of Maxwell and of quantum mechanics without invoking new principles.

That is Lagrangian mechanics can describe in the same formalism and on the conceptual basis both Newtonian mechanics and Maxwell's Equations. The Lagrangians of fields use exactly the same formalism as those of discrete systems (though they have infinite dimensional phase spaces).

If the quote were simply about classical mechanics then any claim of priority for either version would have a hard time because you can arrive at Lagrange's equation starting with nothing more than Newton's Laws (that is without Hamilton's Principle, which is how Lagrange did it; see Goldstein for instance) or at Newton's Laws starting from Lagrange's equation. Thus, the implication of the quote has to be understood in a wider context.

Finally, the application of Lagrange's equation absolutely does not require any foreknowledge at all. The equation is completely local, just like Newton's Laws. The framing in terms of paths minimizing the action simply implies that the rule continues to be applicable as time passes.

Answer 6

For a conservative potential, there is no causality in the least action principle. Feynman has explained this a bit in his The Character of Physical Law Lecture 2 - The Relation of Mathematics to Physics.

[Regarding three equivalent ways to describe the gravitational law: field, action at a distance, and the least action principle] ... Now if you calculate this quantity [Lagrangian] for this route and for another route, you'll get, of course, different numbers for the answer. But there's one route which gives the least possible number for that, and that's the route that the particle takes. Now we're describing the actual motion, the ellipse, by saying something about the whole curve. We have lost the idea of causality, that the particle is here. It sees the pull. It moves to [t]here.

[LAUGHTER]

Instead of that, in some grand fashion, it smells all the curves around here, all the possibilities,

[LAUGHTER]

and decides which one to take. So this is an example of the wide range of beautiful ways of describing nature and that when people talk that nature must have causality, well, you could talk about it this way [LEAST ACTION]. Nature must be stated in terms of a minimum principle. Well, you can talk about it this way [FIELD]. Nature must have a local field. Well, you can do that [ACTION AT A DISTANCE] and so on.

And the question is, which one is right? Now if these various alternatives are mathematically not exactly equivalent and if for certain ones there will be different consequences than for others, then it's a very-- that's perfectly all right then because we have only to do the experiments to find out which way nature actually chooses to do it. Mostly, people come along, and they argue philosophically. They like this one better than that one, but we have learned from much experience that all intuitions about what nature's going to do philosophically fails. It never works. [......]

However, keep in mind that the uniqueness theorem is not always guaranteed. Crazy things can happen when the uniqueness theorem is not held.

For some non-conservative potential, it is possible for the least action principle to give infinite possible paths. Classically, the particle will take one path - out of the many paths - that can only be determined by both initial position and initial momentum. (All of the paths obey the Newtonian equation of motion.) In this case, there is causality in the least action principle (since you really need initial conditions to specify which path is taken, and different initial conditions give different one path). $^\dagger$

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$\dagger$ : Given a "crazy enough" potential field, even Newtonian mechanics turns out to be non-deterministic or not causal.

Answer 7

The potential energy term in the Lagrangian implicitly accounts for forces acting on the body. If it were that in middle of the motion, suddenly new forces were introduced, then a more complicated potential energy function would be required. Hence, everything is consistent.