Re: the energy

[Jack Uretsky <jlu@HEP.ANL.GOV>]

A general remark on this thread.  I think that people are relying too much
on Newton's laws as the foundation of mechanics.  Newton provided only one
of many alternative foundations.  Any argument that relies only on
Newton's laws is bound to be suspect.

Think instead of using variational principles (least action and least
time) formulations of mechanics.  We embed particle motions in a phase
space in which the particle positions and momenta are the coordinates.  A
Lagrangian function incorporates particle interactions ("forces") and a
Hamiltonian ("Energy") functional gives rise to first order equations that
can be solved to obtain paths in phase space.  A slight constraint on the
relationship between positions and momenta (see Dirac's wonderful book)
carries us into quantum mechanics, and relativity can be introduced at the
outset.

        The downside to this way of proceeding is that nobody in its right
mind wants to introduce calculus of variations and functional thinking to
high school freshmen.

                                                Regards,
                                                        Jack


On Fri, 29 Oct 2004, Leigh Palmer wrote:

> On 24-Oct-04, Bob LaMontagne wrote:
>
> > I can stub my toe on a rock and it hurts. I can't stub my toe on
> > gravitational potential energy. I feel very comfortable calling one
> > "real"
> > and one "fictional" or "abstract".
>
> Right on!
>
> and then he wrote:
>
> > Hi All,
> >
> > I'm still getting used to new email software. I unfortunately placed my
> > reply in the wrong location in my last email and my comments may be
> > inadvertently attributed to Paolo. Sorry for the sloppiness.
> >
> > As soon as I sent the last email, I thought of a counterexample to my
> > own
> > comment. If I kick a rock made of Uranium it hurts my toe. If I come
> > back
> > to the same rock after a few million years and kick it again it hurts a
> > "little" less. This is due to internal rearrangement of the particles
> > that
> > make up the Uranium atoms through radioactive decay into other atoms
> > and
> > particles - along with a release in energy as the internal potential
> > energies readjust - resulting in a lower mass.
>
> It is my feeling that you are not directly sensitive to the energy
> here. It is true that the rock is less massive at a later time, but it
> is not what I would call "the same rock". You do indeed perceive the
> mass of the rock in each case, and you can *infer* the energy of the
> rock each time using Einstein's formula. You have an inertia meter of
> sorts in your toe, and because you can perceive it, the mass of the
> rock is real in my meaning of that term. (I note here that I have not
> been reading the discussion on mass.)
>
> On the other hand, I do not believe that there exists an energy meter.
> All measurements of energy are indirect. They are made by measuring the
> perceptible properties of a physical system and applying appropriate
> formulae. That is the lesson in the Feynman parable. That is why I
> classify energy as being more abstract (or less real) than, for
> example, mass.
>
> > It has always seemed a little wondrous to me that a quantity that we
> > introduce as a convenience in Physics 1 to create a scalar way of
> > avoiding
> > the vector nature of Newton's 2nd Law turns out to have some sense of
> > reality in the equation E_0 = mc^2 and also in the General Theory of
> > Relativity.
>
> It is all wonderful, even miraculous, of course.
>
> Leigh

--
"Trust me.  I have a lot of experience at this."
                General Custer's unremembered message to his men,
                just before leading them into the Little Big Horn Valley