Re: Lazy Invariance Question

In article <1149727492.345330.135190@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Edward Green" <spamspamspam3@xxxxxxxxxxx> writes:
mmeron@xxxxxxxxxxxxxxxxxx wrote:

In article <87d5dov25j.fsf@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>, Henning Makholm <henning@xxxxxxxxxxx> writes: ...

In the Newtonian theory, on the other hand, it is not so clear (to me,
at least), how relativity of energy fits together.


By explicitly evaluating the difference in energy as measured relative
to different reference frame. You'll find that if the energy of a
system relative to a refernce frame where its center of mass is at
rest is E, then its energy relative to other reference frame, moving
at constant v relative to the first is

E' = E + (Mv^2)/2

where M is the total mass of the system. Now, since both M and v are
constant, the second term in the above is conserved. Therefore, if E
is conserved, so is E'.

So the key to understanding the invariance of energy conservation in
this case is to wrestle the expression around to this natural
decomposition. If equations could talk, it might be telling us we
really should be working in this special frame, and the remainder of
the energy is a kind of frame artifact.

Well, yes, this is right. The energy of a systme in its CM frame is
"intrinsic". The rest, as you say, is a frame artifact.

How to grok that this magical decomposition is possible? Find a simple
derivation and ponder it, of course. I must try to find one...

Try to write explicitly how does the energy of a system change upon
frame transformations.

I had the (extremely vague) thought that since we had a sum of squares
whose conservation (or in general, whose delta in a particular process,
if not zero) is invariant, we might be looking at a kind of rotation.
Rotations are kind to sums of squares. Of course, we really have a
weighted sum of squares, and I'm very unclear on what would be
rotating. But I am buoyed by the thought that Lorentz transformations
can be thought of as a kind of formal rotation, IIRC, and also your
hint that classical mechanics also has a four vector formulation.
Vague beyond vague.

Well, any connection to rotations is rather accidental here other than
through the fact the we use squares of vectors in both cases. So
(here is another connection) you'll find something very similar to the
result above for moments of inertia. If you take a given body (no
matter what shape and all the possible *parallel (i.e. differing only
by displacement) axes of rotation, there is one around which the
moment of inertia is minimal and the difference betweeen this and the
moment of inertia around any other, parallel, axis is quadratic in the
displacement between the axes. Proven just the same way too. This
statement has a name, somebodies theorem but I don't recall whose.

I also wondered if we might be able to say something clever about
momentum and total energy being the only possible functions enjoying
frame invariant conservation laws (and functions derived from them).
This is also vague beyond vague, but it sound like the kind of thing
clever people might find a way of stating correctly. For example,
suppose we decreed a physics in one frame, which conserved a certain
quantity involving the cubes of velocity. We are only free to enforce
this by decree in one frame, and might find that this quantity was not
conserved in other frames. Has physics worked out so that _only_
quantities allowing frame invariance of conservation laws turned out to
be conserved quantities? Or to put it another way, postulating frame
invariance, does it turn out that precisely everything which could be
conserved subject to this rule is conserved?

Hmm, that's getting complicated. will have to think about it.

Mati Meron | "When you argue with a fool,
meron@xxxxxxxxxxxxxxxxx | chances are he is doing just the same"
.

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