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Re: [Phys-l] rotational dynamics +- straight-line dynamics



On 08/19/2008 03:40 PM, Steve Highland wrote:

I guess what's bothered me over the years is why it is legal to take F=ma
and basically do "r x 'F=ma'" to create the rotational analog for any choice
of r. Is this purely mathematical, or is there some physics in it, too?

What reasoning assigns each force vector (and the ma vector as well) a
particular location in space? What is it that tells us where they act?

Wow. That is quite a deep question.

This is one of those quintessential "phys-l" questions that needs
to be looked at from multiple viewpoints. There are
-- issues at the formal, fundamental physics level, and also
-- issues at the introductory, pedagogical level.

There is some conflict between the levels.

===========

Here is my take on the fundamentals:

As for the assertion that you can derive the rotational laws from the
straight-line laws: I'm not convinced. Asserting that the derivation
is "in all the textbooks" does not impress me; the derivation always
includes some sketchy assumptions.

Here is the counterargument: Conservation of straight-line momentum is
connected (via Noether's theorem) to a particular symmetry in the laws
of physics, namely invariance with respect to an overall translation.
Meanwhile, conservation of angular momentum is connected to a different
symmetry, namely invariance with respect to rotation.

Is anyone bold enough to tell me that invariance with respect to translation
implies invariance with respect to rotation? How much would you like to
bet? It is super-easy to come up with systems that exhibit one symmetry but
not the other. Example: polaroid material. Example: any low-symmetry crystal.

The thing that actually works is the *converse*: If you start with
conservation of angular momentum, it implies conservation of straight-line
momentum. Hint: choose a pivot point infinitely far away.

To repeat: I am not convinced that F=ma is a complete description of
the basic physics, or that the rotational laws are a corollary of the
straight-line laws.

==============

As applied to extended objects (not restricted to point particles), you
can have one or the other of the following notions, not both:
a) Force is a vector
b) F=ma is a complete description of the dynamics.

My preference is to stick with the conventional notion (a). That means
that force, per se, has a direction and magnitude, period.

If you want a usable description of the dynamics, you need something more.
You need a notion that doesn't yet have a name AFAICT. We can call it
an "ultraforce" for now. It has two elements: a plain old force vector,
and a place of attachment.
ultraforce := {force vector, attachment}

Keeping track of the ultraforces is tantamount to keeping track of the
angular momentum, since torque = dL/dt always.
-- It implies keeping track of the forces (and *not* conversely).
-- It implies keeping track of the linear momentum (and *not* conversely).


There's more I could say about this is anybody is interested.