[FRIAM] Least Action

Fri Mar 14 23:21:29 EDT 2025

Let me illustrate my perspective on comparing the Newtonian and Least
Action approaches in the macro world—excluding particle physics, where
quantum mechanics takes over—through a story.

Our protagonists are a proto-engineer and a proto-physicist. Both are
highly intelligent and curious, with a solid grasp of mathematics but no
prior knowledge of the real world. However, they have different goals:
- The engineer seeks knowledge to build things that work.
- The physicist seeks knowledge to deeply understand how the world works.

As they observe bodies in motion, they study kinetic and potential energy
in a world governed by gravity. Both analyze their observations and attempt
to uncover the fundamental laws of nature.

The engineer formulates Newton’s laws of motion. Delighted with their
practicality, he uses them for real-world calculations—designing
trajectories, predicting motion, and solving engineering problems.

The physicist, on the other hand, discovers the principle of Least Action.
He is pleased to have a beautifully simple, overarching law that brings him
closer to a fundamental understanding of nature.

They compare notes and, with mutual respect, conclude:
- Newton’s laws are more suitable for engineering applications.
- The Least Action principle is more fitting for theoretical physics.
- Ultimately, these approaches are not fundamentally different but serve
different purposes—one for practicality, the other for deeper insight.

As for me, Pieter—I wear both hats. Sometimes, I’m an engineer. Sometimes,
I’m a physicist.

On Sat, 15 Mar 2025 at 04:12, Pieter Steenekamp <pieters at randcontrols.co.za>
wrote:

> 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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