(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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