Re: Does Newton's third law hold in Special Relativity?

On Apr 8, 4:48 pm, "Timo A. Nieminen" <t...@xxxxxxxxxxxxxxxxx> wrote:
On Tue, 8 Apr 2008, Darwin123 wrote:
On Apr 7, 10:58 pm, Timo Nieminen <t...@xxxxxxxxxxxxxxxxx> wrote:
On Mon, 7 Apr 2008, Darwin123 wrote:
On Apr 7, 8:37 pm, Timo Nieminen <t...@xxxxxxxxxxxxxxxxx> wrote:
On Mon, 7 Apr 2008, Darwin123 wrote:
On Apr 7, 4:41 pm, "Timo A. Nieminen" <t...@xxxxxxxxxxxxxxxxx> wrote:
On Mon, 7 Apr 2008, ram.rac...@xxxxxxxxx wrote:

Rather, one would use Maxwell's equations (etc) to obtain the same result
as Newton's laws (see further below), in an appropriate limit.

Key qualifier: In the appropriate limit. Maxwell's equations are
still considered more accurate than Newton's Laws. There is no way to
extract a force law consistent with both Maxwell's equations and
Newton's Laws at the same time. As a special limit, sure.

Sure there's a way. See, e.g., Jackson pp 105-108. Once you have the
canonical energy-momentum tensor, you know what the momentum and energy
density of the field is (assuming one can identify the integrands in the
relevant conservation laws with the density of the conserved quantities).
Maxwell's equations tell you how the energy and momentum moves from one
place to another in the field, the Lorentz force tells you the forces
involved in the interaction with matter, and, given that the force
exterted by the field is indeed equal to the rate of loss of momentum of
the field, how can this _not_ be consistent with Newton's laws? Of
course, not consistent with Newton's definition of momentum.

Maybe thats the problem. Newton did not define momentum
explicitly.

He did. First, he defined mass, then "quantity of motion" (the usual
English translation, but it's what we call momentum) as m*v.

He certainly did not come up with "conservation of
momentum." Conservation of momentum can be derived from Newton's Laws
assuming all the mass is contained by "bodies," and that the force is
is invariant with respect to displacement.

Why is this least one needed?
According to Noether's theorem, which applies to both classical
and quantum mechanical systems, every quantity that is conserved has a
corresponding invariant property from which it can be derived. For
example, conservation of energy is associated with invariance in time.
Conservation of a component of momentum is associated with invariance
of a component in position. Conservation of angular momentum is
associated with invariance in angle.
Sometimes the invariance isn't easily measurable on the scale
of a particular experiment.
For example, when the car brakes the tire rubs against the
pavement. Friction turns the kinetic energy of the entire car into
kinetic energy of individual molecules (i.e., heat).
Energy doesn't seem to be conserved because the car slows down.
You can't say the earth absorbs the energy since it is the tire that
heats up. It is hard to see where the invariance to time is. However,
the forces between atoms in the tire are invariant to time.
Momentum does not seem to be conserved since the car slows down.
In this case, it is the earth that absorbs the excess momentum. As
applied to the center of gravity of earth and car, the system is
invariant with respect to the position of the car.

I don't recall an explicit statement of the conservation of momentum in
Principia.
Neither do I. That is my point. Einsteinian relativity is a
modification of Newtonian mechanics, but it is not in any way
equivalent to Newtonian mechanics. Someone on this thread claimed that
the Third Law of Newton was equivalent to conservation of momentum.
But it does not go both ways. The third law of motion gives rise to
the conservation of momentum only under certain conditions. However,
the third law of Newton is not equivalent to conservation of momentum.
Given the laws of optics known in Einsteins time, including both
polarization and radiation pressure, there seems to be no expression
for radiation pressure consistent with the third law of motion. I
think this bothered both Lorentz and Einstein.
The troll was claiming Einstein was stupid. He showed no
originality. He simply used conservation of momentum, and he used it
wrong. Yes, Einstein used conservation of momentum. However, the way
he used it was not obvious to any scientist who understood nineteen
century optics. Apparently, it isn't obvious to everybody even today.
The troll couldn't grasp it even with lots of people showing him.

I think it was Leibnitz who
actually came up with conservation of momentum independent of Newton
and his third law.

I've read very little of Leibniz, but iirc, an important contribution of
his was the recognition of "vis viva", mv^2, as an important quantity in
dynamics, and at least some of argument with Newton/Newtonians was
concerning the relative important of momentum and his almost-KE.

The Leibniz-Clarke (Clarke might have been Newton-by-proxy?)
correspondence might be a good place to look for an early explicit
statement of such conservation laws.
I agree. I may have overextended myself here. I know Leibnitz
discussed kinetic energy in the preEinstein formulation. I jumped to
the conclusion that he made up the conservation laws of both energy
and momentum. Sorry. However, Leibnitz wrote the kinetic energy
formula way before Coriolus. I thought some lurkers may be interested.
Einstein once said that he was trying to resolve the
contradictions created by the existence of radiation pressure. Someone
may well ask what contradictions are in the existence of radiation
pressure. Einstein described a thought experiment that showed clearly
that heat energy, as in the motion of molecules, makes an object more
massive. His analysis presumed that light exerts a pressure, which was
well know at that time. His point was that if radiation exerts a
pressure, than the electromagnetic field has to have inertial mass.

... or, in other language, simply that it must have momentum.
However, the momentum is not in the form p=mv. Not without making
up a rather artificial definition of m and v.
Consider two static fields at right angles to each other. A
static electric field and a static magnetic field. Now, calculate the
flux of energy. Fine! Calculate the flux of momentum. Fine! See, they
are both nonzero. However, what exactly is moving with a nonzero
velocity? In other words, where is the body?

The classic treatment of the transport of energy by fields (in the general
sense, so including stresses in elastic bodies etc) and the associated
momentum was by N. A. Umov, c. 1874, but little known outside Russia. The
essential result is that the momentum flux p is related to the power P by
p=P/v, where v is the speed of energy transport.
That looks artificial to me. Not that I am putting down Umov. Do
you see the lengths physicists were going to preserve p=mv?

SR provides a neat unification of this result with the transport of energy
and momentum by moving bodies.
Yes. Very neat.

Hmm - I didn't read the troll-end of the thread (and I don't see much
point doing so).
Good for you. I just wanted to have a conversation with someone
like you, anyway. I wasn't answering the troll. I was knocking down a
straw man model just for attention. And because I keep meeting people
like that. After a run in with a goof like that, I always wish I had
said something before they embarrassed someone else.
.

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