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On 02/20/2008 10:07 AM, LaMontagne, Bob wrote:
> I think I am confused here for the same reasons. My understanding is
> that the game here is to demonstrate that a conservative force can
> increase the TOTAL energy of a system (not just simply do work). I am
> having a hard time seeing how the example presented can be an isolated
> system.
Isolated???
The question I was addressing was *not* restricted to isolated
systems. Please let's not move the goal posts during the game.
Also: I do not *want* to restrict it to isolated systems. It
is really quite important that the law of conservation of energy
(and momentum, and charge, and lepton number, and ....) be applicable
to non-isolated systems and sub-systems.
=====================================================
On 02/20/2008 11:02 AM, Steve Highland wrote:
>> Now I am beginning to catch on to how our usages of the term "conservative
>> system" differ.
:-)
>> You seem to be calling a system conservative if there's a way to track the
>> energy flowing across the system boundary and matching it with the change in
>> energy inside the system.
Yes.
>> I am calling a system conservative if there is a way to
>> define a total energy for the system that stays constant as the system
>> evolves.
That short-changes the notion of "conservation". If we're talking
about global conservation and/or an isolated system, conservation is
the same as constancy. But when it comes time to actually apply the
conservation law, it is really quite important to have a *local* law,
applicable to non-isolated systems and sub-systems.
>>> Conservation of XX means that XX is constant _except_
>> insofar as XX flows
>>> across the boundary of the system." [John Denker]
:-)
>> I understand the idea of local conservation just fine when dealing with
>> conservation of electric charge. Charge strikes me as real 'stuff' that
>> moves from place to place, so it can't just pop out of existence in one
>> place and appear someplace else.
OK!
>> Energy does not seem the same to me. I tend to go back and forth on whether
>> energy is 'real' or just bookkeeping, but right now I'm going the
>> bookkeeping route.
To me, conservation of energy is on exactly the same footing as
conservation of charge. Energy is somewhat abstract, but so is
charge. Energy can be viewed as "stuff" that can flow across
the boundary, and so can charge. Energy can be formalized as a
completely abstract bookkeeping exercise, and so can charge.
>> Going back to the example of a ball falling in a gravitational field, how do
>> you locate the energy flow into the system of just the ball? I don't see
>> how to identify an energy flux across any boundary. If there is one I would
>> think that the flow must get denser near the ball, because a mathematical
>> surface enclosing the ball close to the ball's surface has a smaller area
>> than one enclosing the ball along with some air around it.
>>
>> Unless there's some way of creating something along the lines of a Poynting
>> vector for gravitational fields I'm lost here.
Poynting is the right answer. Classical gravitation looks a whole
lot like electrostatics, and you can define a gravitational field
energy in analogy to the electrostatic field energy. It's kinda
creepy because is has the opposite sign, but it works OK within the
classical approximation.
This is why the notion of "field" was invented. It allows conservation
laws to be enforced locally. This is a big improvement over action at
a distance.
Tangential remark: If you want a fully relativistic approach to
this problem, both the question and the answer become considerably
more complicated ... but there *is* a local conservation law for
the stress-energy tensor.
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