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

[David Bowman <David_Bowman@georgetowncollege.edu>]

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