I think having an answer that’s “right” that everyone can agree with is rare in our culture, though it does seem to happen a lot in math and physics.

We would say a student is thinking critically when they look at a research source and critically evaluate the information in it. Were there limitations to the equipment used that weren’t noted? Was there a different system that could have been studied to make the result more general? Are the conclusions statistically significant? What’s interesting is that these questions actually have answers, though it’s possible neither the student nor that author knows that answer (yet).

]]>Of course, I don’t see how this would take place in either mathematics or physics (I can see how team teaching could be useful, but not as a tool to get students to understand critical thinking in this way).

Do you think that mathematics and science could be part of the problem? In our fields, it seems like critical thinkers either (1) have to agree if there is a proof/enough data or (2) can disagree when there is not yet a proof/is not yet enough data. Maybe this is a consequence of our society being too scientifically literate (probably not).

]]>I wonder if the adiabatic invariants of the Kepler problem would be useful in getting some kind of analytic expression for the radial coordinate as a function of time, if only in the limit where the mass drops extremely slowly. I think the formal condition in this context amounts to the fractional rate of change of mass being much less than the frequency of a single orbit. Seems like that should be the case for the sun and any given planet.

If I remember correctly, there should be two independent action variables, for this system which should remain approximately conserved if the change to the Hamiltonian is slow enough to obey that condition above. The first invariant is just the angular momentum, but the radial invariant is probably less trivial. The example I always remember is calculating the period of a simple pendulum whose length is gradually changing. I’m more than a little rusty with action-angle variables, but I’ll trying playing around with this!

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