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Re: [Phys-L] let's define energy



   http://www.av8n.com/physics/thermo/energy.html . This is a good site. But I want to suggest two corrections.
1)  In Sec. 1.3 there are 2 equations for kinetic energy K in terms of p and v:                     K=(1/2)pv   at low speeds        (1.2f)     and              K= pv      at high speeds       (1.2g)   The exact expression for kinetic energy is K=E-mc^2 = ((p/v)-m)c^2 = (gamma(v)-1)mc^2. It reduces to (1.2f) at low speeds. But at high speeds it is approximated by K--->mc^2 gamma(v)=(p/v)c^2. It coincides with (1.2g) only at v=c (when p is infinite for any massive particle). But analytically, it is quite different from (1.2g) at all v other than c, and I expect it to be more accurate numerically. 
2) In the beginning of Sec. 1.5 we read:  "...suppose an electron meets a positron. The two of themannihilate each other, and a couple of gamma rays go flying off, with 511 keVof energy apiece. In this situation the number of electrons is not conserved (true!),the number of positrons is conserved (false!), and mass is not conserved (false!). However,energy is conserved (true!)." (Notations in italics are mine - MF)   The first falsehood here is an obvious typo. But the second one may represent a widely-spread (and fashionable!) misconception confusing non-additivity of the rest mass with its non-conservation. The latter would contradict the very tenet of the discussed site: non-conservation of the rest mass would mean non-conservation of the rest energy of the system.   The corollary of this argument: even though the rest mass of a single photon with definite momentum is zero, the rest mass of the two photons with non-parallel momenta is not. In the considered case, the pair of the produced photons has the center of mass, and its rest mass is equal to 2 X 511 keV /c^2.    I will not go into farther details, since this topic has been extensively discussed on this Forum without any agreement.     Moses Fayngold   NJIT




On Saturday, September 26, 2015 10:26 PM, David Bowman <David_Bowman@georgetowncollege.edu> wrote:


Regarding definitions of energy:

....

2) How many different definitions of /physics/ energy are
we talking about?  Is there more than one serious contender?

I've been wrong about this sort of thing before, but I
would hope the community could come to a consensus on
how to define the /physics/ energy.

  This stands in contrast to things like:
    -- "adiabatic", where there are two long-established
    meanings, neither of which is particularly better or
    worse than the other.
    -- "heat", where there are at least four long-established
    and widely-used meanings, each of which has some merit
    but also some serious problems.  (Not to mention various
    vernacular and/or metaphorical usages.)
    -- etc. etc. etc.

3) Within "science" broadly, I know of two or three definitions
of «energy» ... only one of which is the /physics/ energy.

a) The /physics/ energy, as I understand it, is unique, well
  defined, and well behaved.  Here's how I explain it:
      https://www.av8n.com/physics/thermo/energy.html
  or equivalently
    http://www.av8n.com/physics/thermo/energy.html

b) Meanwhile, there is also the Department of Energy «energy».
  This involves some notion of "available" or "useful" energy.
  This is important, but it's not the /physics/ energy.
  Definitely not.  When the DoE says "please «conserve» «energy»"
  they are not using the physics notion of energy *or* the
  physics notion of conservation.

c) In dictionaries and in third-grade science books you often
  see energy defined as "the ability to do work".  This is a
  rough approximation to the DoE «energy».  It is absolutely
  not the physics energy.  For details on this, see
  https://www.av8n.com/physics/thermo/energy.html#sec-workability

....

  2) Does anybody know of any other viable, useful, or even
  plausible ways of defining the /physics/ energy?

....

The definition of energy that I've kind of been partial to is:

Energy: The numerical value of that expression that generates an infinitesimal displacement of the state of an isolated dynamical system in time.

This definition seems to work for both quantum and classical dynamical systems as long as the term "numerical value" is understood to be an element of the spectrum or eigenvalue of the Hamiltonian operator in the quantum case.  In a thermodynamical situation one would take the macroscopic average of this numerical value whose precise exact value defined in terms of the microscopic degrees of freedom would be considered as a statistical random variable because of the state being defined in terms of a density matrix or phase space distribution function with a nonzero entropy.  Because of the dependence of the expression generating infinitesimal temporal displacements (i.e. the Hamiltonian function/operator) on the coordinate system of the frame of reference used the value of the energy is also frame dependent according to the canons of how frame-changing transformations are to be done, whether being Galilean or Lorentz transfomations in situations flat space-time and being resticted to only inertial frames, or more general coordinate coordinate transformations in more general situations (such as between noninertial frames, or even in the curved spacetime situations of GR).

David Bowman
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