atheorist (atheorist) wrote,

Galileo had an argument against Aristotle's law of gravity.
Aristotle's law of gravity was that objects fall at a speed proportional to their weight.
Galileo's argument against it was something like:
"Consider two things connected together only loosely. On the one hand, considering them as the aggregate thing, they should fall fast. On the other hand, considering them as two separate things, they should fall slow. What would the tension in the last strand of twine be like? This is weird."

This is a kind of argument from continuity - Aristotle's law has a discontinuity as you go from a single barbell-shaped object to two adjacent objects nearly touching. If we believe that the laws of nature ought to be continuous with respect to that transformation, then we can reject Aristotle's law from the armchair.

One thing you can do with a circuit is draw its signal flow graph. For some simple circuits, drawing the signal flow graph is follow-your-nose easy. For some very slightly more complicated circuits, you get arguments like this:

Consider a current source (Sf) wired up in parallel with two other branches. The first branch has a resistor (R1) and a voltage source (Se1), while the second branch is similar (R2, Se2).

Trying to draw the signal flow graph (in the time domain), we might say:
1. Start at the current source, Sf.
2. Let x be the current through the 1 branch. (Note, we could have gone the other way).
3. Then Sf-x is the current through the 2 branch.
4. So (Sf-x)*R2 is voltage across R2.
5. So (Sf-x)*R2+Se2 is the voltage across the whole circuit.
6. So (Sf-x)*R2+Se2-Se1 is the voltage across R1.
7. So ((Sf-x)*R2+Se2-Se1)/R1 is the current through the 1 branch, that is, x.
8. Solving for x, we find that x==(Sf*R2+Se2-Se1)/(R1+R2).

(This is called an algebraic loop in bond graph terms).

This process of predicting what the circuit will do is not "shaped like the circuit". It involves steps that are contingent on feeling stuck, it has asymmetries where the circuit has symmetries, it's nonmechanical. We could make the final symbol-juggling almost arbitrarily hard by introducing nonlinearities, but apparently the circuit can juggle those symbols essentially instantaneously. This cannot be how the circuit itself computes its behavior.

Perhaps it was foolish of me to believe that the human process of solving the easy circuits was analogous to the circuits' method of computing its behavior.

If we spread the circuit far enough apart, the connections between the pieces will need to be modeled with transmission lines. In order for the constitutive laws to be continuous with respect to whether we model the circuit as containing transmission lines or not, the transients in the transmission-line variant of the circuit ought to die out quickly. Furthermore, the transients can probably be viewed as computing the answer to the set of constitutive equations, perhaps by iterative relaxation (Jacobi or Gauss-Seidel methods?)

I think this might be a 7th or 14th order differential equation, depending on how you model transmission lines (or whether you count a complex number as one or two degrees of freedom)? Regardless, it probably converges toward a steady state pretty rapidly in a lot of reasonable models of transmission lines.

That is, the circuit laws don't care exactly how you parse the circuit, because they're continuous with respect to "nearly the same" parses, even though I personally feel more confident in solving a single linear equation than computing the steady-state behavior of a differential equation.

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