<div dir="ltr">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. <br><br>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: <br>- The engineer seeks knowledge to build things that work. <br>- The physicist seeks knowledge to deeply understand how the world works. <br><br>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. <br><br>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. <br><br>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. <br><br>They compare notes and, with mutual respect, conclude: <br>- Newton’s laws are more suitable for engineering applications. <br>- The Least Action principle is more fitting for theoretical physics. <br>- Ultimately, these approaches are not fundamentally different but serve different purposes—one for practicality, the other for deeper insight. <br><br>As for me, Pieter—I wear both hats. Sometimes, I’m an engineer. Sometimes, I’m a physicist.</div><br><div class="gmail_quote gmail_quote_container"><div dir="ltr" class="gmail_attr">On Sat, 15 Mar 2025 at 04:12, Pieter Steenekamp <<a href="mailto:pieters@randcontrols.co.za">pieters@randcontrols.co.za</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr">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.</div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, 14 Mar 2025 at 15:23, Barry MacKichan <<a href="mailto:barry.mackichan@mackichan.com" target="_blank">barry.mackichan@mackichan.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><u></u>
<div><div style="font-family:sans-serif"><div style="white-space:normal"><p dir="auto">(Sorry about the previous message: the key that I thought typed ∆ actually sent the email before I finished. As I was saying:</p>
<p dir="auto">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?</p>
<p dir="auto">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.</p>
<p dir="auto">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.”</p>
<p dir="auto">— Barry</p>
<p dir="auto">On 14 Mar 2025, at 9:08, Barry MacKichan wrote:</p>
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<p dir="auto">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</p>
<p dir="auto">On 12 Mar 2025, at 11:44, Pieter Steenekamp wrote:</p>
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<div dir="ltr">There's a *"nice"* layman’s explanation of the principle of *least action* (<a href="https://www.youtube.com/watch?v=qJZ1Ez28C-A)" target="_blank">https://www.youtube.com/watch?v=qJZ1Ez28C-A)</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.) <br>
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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. <br>
<br>
A common example used to illustrate this principle goes like this: <br>
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. <br>
<br>
This framing presents a problem. Here’s my perspective: <br>
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. <br>
<br>
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. <br>
<br>
I don’t buy that. <br>
<br>
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. <br>
<br>
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.<br>
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