What is the most difficult concept to grasp in physics? originally appeared on Quora: the knowledge sharing network where compelling questions are answered by people with unique insights.
Answer by Alejandro Jenkins, Caltech PhD '06 (physics), Harvard AB '01 (physics & math), on Quora:
A famous Harvard physics professor (Ed Purcell maybe?) said that undergraduate physics students come in expecting that the hardest thing they'll have to learn will be either relativity or quantum mechanics. Actually, those are the most novel topics (i.e., the ones involving notions that are the most surprising from our ordinary, common-sense perspective). The hardest thing that an undergraduate physics students must learn is the classical dynamics of spinning tops (also called "rigid bodies" in this context).
Having taught classical mechanics to advanced undergraduates in physics, I find this to be true. The following figure, which I've taken from chapter VI, sec. 37 of the Mechanics by Landau and Lifshitz, shows possible values of the angular momentum vector, in the non-inertial body frame, for a free, asymmetric top. The ellipsoid is a surface of constant energy, and the closed curves are given by the intersection of that ellipsoid with spheres of various radii, corresponding to different values of the total magnitude of the angular momentum:
If you still don't believe me that tops can really be such a headache, I suggest looking up the "Poinsot construction," which even inspired a humorous poem by Professor David N. Williams of the University of Michigan. Or check out the explicit solutions to the motion of the free asymmetric top in terms of Jacobi elliptic functions (and this is for a free top, mind you, with no net torque acting on it).
The great theoretical physicist James Clerk Maxwell (1831-1879), discoverer of the laws of electrodynamics, wrote, "To those who study the progress of exact science, the common spinning-top is a symbol of the labours and the perplexities of men."
Actually, tops can be such a tricky subject to teach that many lecturers tend to gloss over them, especially now that we're in a rush to get to quantum physics. Still, even though the classical mechanics of spinning tops can be hard to grasp, it's perfectly well defined. The mathematics of (non-relativistic) quantum mechanics is fairly straightforward by comparison, but the interpretation of what the rules of quantum mechanics mean, especially insofar as they concern the process of measurement, remains quite obscure. Most physicists are content to compute observable quantities, leaving the interpretation to the philosophers, an attitude captured in a famous dictum often wrongly attributed to Richard Feynman, to "shut up and calculate;" see N. D. Mermin, "Could Feynman have said this?", Physics Today 57, 10 (2004).
Things do get pretty hairy when you need a description that's both quantum and relativistic, which is the regime of high-energy physics. This requires what's known as quantum field theory, which is a subject that still presents many conceptual difficulties despite its great predictive successes. But quantum field theory is not usually studied at the undergraduate level.
A friend recently pointed out to me this video of a T-handle spinning at zero gravity in the International Space Station, which provides an even more striking demonstration of the tennis racket theorem.
The T-handle is initially spun about the principal axis with intermediate moment of inertia, which is not stable. The result is that the axis reverses repeatedly. This particular behavior has recently been dubbed the "Dzhanibekov effect" after the Soviet cosmonaut Vladimir Dzhanibekov, who demonstrated it in space in 1985 (though the relevant physics has been understood at least since the work of Euler in the mid-18th century).
This gives me another opportunity to reflect on some of the headaches associated with understanding the physics of spinning tops. There's no torque acting on the T-handle, so its angular momentum ought to be fixed. How can this be reconciled with the fact that the axis of rotation is repeatedly flipping? The answer is that it flips in the top's own frame of reference (which is not inertial). If you examine the video carefully you can see that, with respect to you as an inertial observer, the T-handle is spinning in the same direction before and after the flip!
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