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Re: [Phys-l] momentum conservation ==> action=reaction



Doesn't your step 4 essentially declare N2? Agreed that your eqn [1] leads to your eqn [2]. But it is another matter altogether to assume F = (d/dt)p. That seems either equivalent to accepting N2 or requiring a set of experiments to demonstrate its validity.

Bob at PC

________________________________

From: phys-l-bounces@carnot.physics.buffalo.edu on behalf of John Denker
Sent: Fri 8/1/2008 3:12 PM
To: Forum for Physics Educators
Subject: [Phys-l] momentum conservation ==> action=reaction



1) Start with a strict local law of conservation of momentum.
Express the law the same way you would express any other
conservative flow, i.e.

(d/dt)(p)(inside boundary) = flow(p)(across boundary) [1]

For details on this, see
http://www.av8n.com/physics/conservative-flow.htm

2) For simplicity, restrict attention to a scenario where there
are just two distinct subsystems. (For further simplicity, the
subsystems may optionally be considered _particles_, but this
is not actually necessary.)

3) The flow terms are equal and opposite, because what flows
outward across boundary 1 must flow inward across boundary 2.
Proof by exhaustion: there is nowhere else for the momentum
to go.

Therefore by equation [1] we have
(d/dt) p1 = - (d/dt) p2 [2]

4) Define force to be (d/dt) p. Conclude from [2] that the
forces are equal and opposite.

Note that we have not invoked Newton's first or second law, just
the third.

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

In the reverse direction, action=reaction is /not/ sufficient to
prove strict local conservation of momentum.

*) Minor point: conservation applies to all types of interactions,
not just pairwise interactions, so you would need to consider a
richer set of scenarios, with N subsystems and on the order of
N^2 pairwise interactions.

*) The fatal limitation is that the law of equal-and-opposite
forces can be (and often is) applied in action-at-a-distance
situations, such as Newtonian gravity, that violate _local_
conservation of momentum.



On 08/01/2008 08:55 AM, Dan L. MacIsaac wrote:
N3: F_12 = - F_21
..........
regroup, et voila:
dP_1 / dt + dP_2 / dt = 0 = d(P total) / dt

conservation of linear momentum for the two interacting objects

That shows that action=reaction is /consistent/ with conservation in
a /special case/ but does not prove the general proposition.

Let's be clear: N3 is a subset of local conservation of momentum.
It's a /proper/ subset. Be careful not to reverse the direction of
the "==>" sign.

A conservation law that is not /local/ is
a) not nearly so useful, and
b) inconsistent with relativistic causality.

For details, see:
http://www.av8n.com/physics/conservative-flow.htm



On 08/01/2008 08:43 AM, David Whitbeck wrote:

.... momentum conservation is a Noether Current associated with
spatial translation invariance. Spatial translation invariance is a
very intuitive, natural and easy to swallow axiom. There's really
nothing strange or mystical about momentum conversation, it's simply
a reflection of symmetry.

That's true and important.

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