[FRIAM] Least Action

Fri Mar 14 22:12:09 EDT 2025

While the principle of least action is a powerful tool for calculating the
path of a physical system, it’s important to understand that it doesn’t
imply that the system has any foresight about its future position. The
ball, when thrown, doesn’t “know” where it will land; it simply follows the
path determined by its initial conditions and the forces acting on it. The
principle of least action is a mathematical description that happens to
characterize that path, not a mechanism by which the ball plans its
trajectory. Popular science might sometimes personify the ball, suggesting
it “chooses” the path of least action, but that’s as accurate as saying
that a rock chooses to fall when dropped. The ball is just acting according
to the laws of physics, not planning its actions.

On Fri, 14 Mar 2025 at 15:23, Barry MacKichan <barry.mackichan at mackichan.com>
wrote:

> (Sorry about the previous message: the key that I thought typed ∆ actually
> sent the email before I finished. As I was saying:
>
> I don’t see that there is any problem here. Suppose at some point the ball
> reasons as follows: I’ve gotten to this point, and my trajectory so far is
> the one with the least action. What is the vector I should follow for the
> next ∆t time interval so that my path continues to have the least action?
>
> The answer to that question (although I haven’t worked it out recently)
> must be that the motion has to satisfy the differential equations that the
> ball is computing.
>
> In other words, the theorems are “All solutions of these equations have
> this property” and “all trajectories that have this property must satisfy
> these equations.”
>
> — Barry
>
> On 14 Mar 2025, at 9:08, Barry MacKichan wrote:
>
> I don’t see that there is any problem here. Suppose at some point the ball
> reasons as follows: I’ve gotten to this point, and my trajectory so far is
> the one with the least action. What is the vector I should follow for the
> next
>
> On 12 Mar 2025, at 11:44, Pieter Steenekamp wrote:
>
> There's a *"nice"* layman’s explanation of the principle of *least action*
> (https://www.youtube.com/watch?v=qJZ1Ez28C-A)—though I don’t quite agree
> with it. (It does, however, include a rather neat explanation of quantum
> mechanics that I find useful—but that’s another discussion.)
>
> Back in engineering school, when calculating trajectories, we relied
> entirely on Newtonian mechanics, applying it so relentlessly in
> problem-solving that it became second nature. Later, I encountered the
> principle of *least action* and its claim to be more fundamental than
> Newton’s laws.
>
> A common example used to illustrate this principle goes like this:
> If someone throws a ball from point A to point B, the ball *evaluates* all
> possible paths and then follows the one of least action.
>
> This framing presents a problem. Here’s my perspective:
> If a person throws a ball from point A and it *happens* to land at point
> B, a post-mortem analysis will confirm that it followed the path of least
> action. But that’s an observation, not a mechanism.
>
> The distinction is subtle but important. In both cases, when the ball
> leaves the thrower’s hand, it has no knowledge of where it will land. Throw
> a thousand balls with slightly different angles and velocities, and they’ll
> land in a distribution around B. Yet the layman’s explanation suggests that
> each ball somehow *knows* its endpoint in advance and selects the
> least-action trajectory accordingly.
>
> I don’t buy that.
>
> My view (and I welcome correction) is that the ball simply follows
> Newton’s laws (or the least action laws) step by step. It doesn’t *choose*
> a trajectory—it merely responds to the local forces acting on it at every
> instant. Once it reaches its final position, we can look back and confirm
> that it followed the least-action path, but that’s a retrospective
> conclusion, not a guiding principle.
>
> Ultimately, in this context, Newton’s laws and the least-action principle
> are equivalent descriptions of the same physics—neither requires the system
> to "know" its endpoint in advance.
>
>
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