On 7/21/21 12:59 AM, bernard cleyet wrote in part:
The ODE doesn’t care how the pendulum is driven.
The physics cares! If the ODE can't express this, then it's
the wrong ODE.
There is some magnificent physics here. Let's work through
it.
Short version: Starting with an equation of motion in terms
of position, then using nothing but high-school algebra, you
can rewrite it in terms of momentum. It's the same equation.
The first equation has dimensions of force, and it may seem
"natural" to drive it using such-and-such specified force
... while the second equation has dimensions of velocity,
and it may seem "natural" to drive it using such-and-such
specified velocity ... but both equations represent the same
physics. It's just a change of variable. Different driving
schemes give different results. The physics does not care
whether you think this is "natural" or not.
ALSO (!!!!!) there is parametric drive. This is quite
distinct from either of the drive schemes mentioned above.
It's fascinating and very powerful. I do not have time right
now to explain it.
If you aren't familiar with this, I vehemently recommend you
do the experiment. The taller the swing-set, the better.
The electrical notation is more familiar to most people, so
let's examine that ... but it must be emphasized that the
same goes for any other oscillator: the same principles, the
same techniques, the same conclusions.
For a series LC, at resonance the impedance of the series
combination is zero. If you drive it with constant voltage
you get infinite current. Meanwhile, for a parallel LC, at
resonance the impedance of the parallel combination is
infinite. If you drive it with constant current you get
infinite voltage.
The series circuit with zero drive voltage is *exactly* the
same circuit as the parallel circuit with zero drive
current. Draw the diagram! The only difference between the
two circuits is the drive scheme. The drive scheme matters
tremendously .
Forsooth, you can have a *single* circuit that contains two
drivers, constant-voltage and constant-current. It contains
the usual "series LC" and "parallel LC" as special cases.
The drive scheme matters tremendously.
=============================
Long version. Serious physics.
Let's start by writing down the Lagrangian. When in doubt,
that should almost always be the first step. The Lagrangian
knows all and tells all.
The grade-school notion of the capacitor energy and inductor
energy are ½CV² and ½LI² respectively ... but that uses two
different variables. Let's pick charge (Q) as our variable.
Then for a capacitor V = Q/C (by definition of C) and in all
generality I = Qᣟ (by definition of I). So that gives us
L = KE - PE
= ½LQᣟ² - ½Q²/C
where
Qᣟ ≡ Q dot
≡ dQ/dt
Note that the PE depends on Q while the KE depends on Qᣟ, as
they should.
Turning the crank in the usual Euler-Lagrange way, this
gives us the equation of motion:
LQᣟ + Q/C = 0
which has dimensions of voltage. If you add a driving term
on the RHS it must be a low-impedance /constant-voltage/
source.
HOWEVER ... we can do the same calculation again, picking a
different variable. The formalism does not care what you
choose as "the" variable. In particular, instead of charge
we could choose flux.
φ = flux
= LI (by definition of L)
V = φᣟ (Maxwell equation)
So the Lagrangian is:
L = KE - PE
= ½C φᣟ^2 - ½φ²/L
It must be emphasized that except for an overall minus sign,
this is the *same* Lagrangian ... and the variational
principle (δL = 0) does not care about an overall minus
sign.
Turning the Euler-Lagrange crank again, we get the equation
of motion:
Cφᣟ + φ/L = 0
which has dimensions of current. If you add a driving term
on the RHS it has to be a high-impedance /constant-current/
source.
/////////////////////////
Tangent:
Let's go back to picking Q as our variable. Then basic
Euler-Lagrange theory tells us that the the conjugate
momentum is
momentum = ∂L/∂Qᣟ
= LI
= φ (i.e. the flux)
It is worth remembering that /charge/ and /flux/ are
dynamically conjugate. This means (among *many* other
things) that there is a Heisenberg commutation relation:
[Q, φ] = i ℏ
Similarly, if you pick φ as "the" variable, then you find
that -Q is the dynamically conjugate momentum. For this
reason, charge and flux are much more convenient than
voltage or current. Charge is CV and flux is LI, so there is
a close connection, but the physics is much easier to see in
terms of Q and φ.
Note that Q and φ are 90° apart in phase space. More
generally, you can pick some random direction in phase space
(some combination of Q and φ) as "the" coordinate, and the
formalism will find the appropriate momentum. You can look
this up in books on classical mechanics under the heading of
"contact transformation".
Bogoliubov transformations are similar.
///////////////////////
History and sociology note:
When I was in school, I discovered there were certain folks
who had taken classical mechanics, but weren't content to do
the textbook exercises. To test their understanding, and to
see if the theory made any sense, they took the theory for a
mass on a spring and applied to an electrical LC.
Similarly, they used standing waves on a piano string as an
example of field theory.
The folks who have gone through these rites of passage tend
to find each other. Folks who are curious about practical
applications of fundamental physics.
Also: In case you were wondering how Nyquist derived the
formula for the noise in a resistor, it was a direct
application of thermodynamics and field theory to an
electrical circuit. Getting the idea to try that was
brilliant. Once you get the idea, the calculation is short
enough to fit into a Phys Rev Letter. For extra style
points, his buddy Johnson did the experiment, and they
publish the two Phys Rev Letters back-to-back. This was
in 1928.