[FRIAM] Least Action

[Barry MacKichan]

(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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