[Phys-L] the same equations have the same solutions

[John Denker <jsd@av8n.com>]

On 5/5/21 4:00 AM, Carl Mungan wrote:

> 2. I always thought of permeability being to magnets what
> permittivity is to dielectrics. But now instead permeability is being
> made analogous to conductivity. Does permittivity likewise map onto
> some kind of analog of conductivity?

All of the above. AFAICT there is not any deep issue here.

There are lots of different analogies.  The existence of
one mapping does not rule out another mapping, or really
affect it in any way.

There are lots of situations where you can have series
and parallel combinations, which invite analogies to
Ohm's law. For example, the topic of /security/ involves
series and parallel combinations. Increasing the thickness
of your front door might add security (in series) but not
if the side door is standing open (in parallel).

We can really drive home this point with a physics example.
It's obvious that an electrical LC oscillator is deeply
analogous to a mechanical mass-on-a-spring oscillator. But
the question arises: what is "the" mapping:

 a) L <=> mass
    C <=> spring
    q <=> position

 b) C <=> mass
    L <=> spring
    φ <=> position

The answer is that either one works perfectly well. In the
language of classical mechanics, the two mappings differ by
a contact transformation.

Note that the Lagrangian is the same in either case (within
an unimportant minus sign). So the variational principle
	δL = 0
is exactly the same in either case. Pick a variable (e.g. q
or φ) and turn the crank. Euler-Lagrange blah blah.

q and φ are the most elegant choices, but other choices for
the "position" variable are allowed also.

I really recommend doing this calculation. I know a lot of
physicists who vividly remember doing this the first time.
It's one of those "Wow, physics actually works" moments.
It's a rite of passage.

And by the way, q and φ are dynamically conjugate. This comes
in handy if you are labeling the axes in phase space ... or
writing down the Heisenberg uncertainty principle in electrical
units.

While we're on the topic of "the same equations have the same
solutions" and mechanical analogies: I know a lot of physicists
who have worked out the physics of a piano string using the
methods of /field theory/. Sum over modes and all that. This
another of those "rite of passage" moments.