Re: [Phys-L] Again: Re: causality

[John Denker <jsd@av8n.com>]

In the context of F=ma, I have often asserted that
  force does not /cause/ acceleration
and by the same token
  acceleration does not /cause/ force.

The 2nd law says you cannot have one without the other,
and that they happen at the same time;  that is:
  F(t) = m a(t)

On 4/11/19 12:53 PM, bernard cleyet wrote:
> I’m not certain all this happens simultaneously..  If one “pushes”
> against the side of a brick one compress the first plane of molecules
> with respect with next one, so maybe that’s instantaneous, and then
> the next plane is compressed so what one is a longitudinal wave of
> motion.  The whole brick doesn’t move immediately; no?

That's an interesting and fairly common question.
To make progress, we need to wash away the imprecise
English-language description and focus on the precise
physics.

In particular, it is imprecise and misleading to talk
about "the" force and "the" acceleration of the brick.
In fact there are lots of forces and lots of acceleration.

One of the biggest swindles that is foisted on students
in the introductory course is this:  Newton's laws are
presented in their simplest form, which strictly speaking
applies only to /pointlike/ particles.  Subject to some
restrictions, the laws can be extended to rigid bodies,
but they become more complicated (and more restricted!)
in the process.  They can further be extended to media
with internal degrees of freedom, but they become *much*
more complicated in the process.

Applying the simple F=ma law to a brick violates crucial
restrictions if you look at the very short time-scales
and length-scales called for in the question.  Bricks
are proverbially rigid, but they are not *actually*
rigid if you look so very closely.  So the English is
not just imprecise, it is aggressively misleading.

Even in the lowest-level calculus-based course, you can
model a brick as a lattice of pointlike particles linked
by massless springs.  You can apply Newton's laws in their
simplest form on a point by point basis, and thereby
derive the sound wave equation.

The solutions exhibit a strong form of causality: effects
must come /later/ than causes, by an amount

	Δt ≥ Δx / c

where c is the speed of sound.  This is a /cone/ of causality,
analogous to the light-cone in special relativity.

Finally, to answer the main question that was asked: This
*still* does not change the symmetrical relationship between
force and acceleration.  At any particular place:
 -- they still happen at the same time, and
 -- you still can't have either one without the other.

Let's be clear: For any selected set of brick-particles:
 *) You can apply a known force and infer the acceleration.
... OR ...
 *) You can apply a known acceleration and infer the force.

Inference is not the same as causation.

The presence of other particles makes the inferences much
more complicated, but doesn't change the essential symmetry
of the relationship between force and acceleration.

You don't need to take my word for this.  You can actually
prove it, although the simplest proof I know requires more
machinery than is available in the introductory course, and
is more suited to a upper-division course for physics majors,
i.e. formal classical mechanics.  Replace "force" by dp/dt
in everything I just said.  If you know the Lagrangian, you
can choose almost anything you want as the coordinate, and
then the Lagrangian will tell you the appropriate conjugate
momentum.
	"The Lagrangian knows all and tells all."
	(In contrast, the Hamiltonian does not.)
You can even go nuts and choose p as your "coordinate" in
which case -x becomes the appropriate dynamically conjugate
momentum.  That's weird, and the terminology breaks down,
but the math works fine.  This can be thought of as rotating
your axes 90° in phase space.  (You can rotate by any angle
you like, and still describe the same physics.)  All these
are examples of canonical transformations aka Bogoliubov
transformations.  They leave area in phase space invariant,
which is a good sign, required if you want to be consistent
with the second law of thermodynamics and the Heisenberg
uncertainty principle and the optical brightness theorem
and Liouville's theorem (which are all essentially the same
thing).

The point remains:  Since you are formally guaranteed that
you can turn x into p and p into -x, there is absolutely no
chance that force "causes" acceleration (or vice versa).