The "shuddering" singularity of Euler's disk is the best example I can think of. As a prototype of mechanical behavior with a "finite time singularity", a paper by Moffatt in Nature made quite a splash in 2000. However, there were a number of problems with both the physics and the math in that paper. A readable summary of the issues can be found here: http://ruina.tam.cornell.edu/research/topics/miscellaneous/comments_on_moffa... If more generally you are interested in examples of physical dynamics, where regular initial conditions lead to singularities (that are resolved in the breakdown of the physical model in extreme limits) in finite time, one of the most famous is the regularity of the solutions of the Euler equations for incompressible, inviscid fluids. It's believed that you can get singular vortex cores in a finite time, but last time I checked there was no proof. -Veit On Dec 18, 2013, at 3:24 PM, James Propp <jamespropp@gmail.com> wrote:

For years I've taught my honors calculus students about functions like x^2 sin 1/x, and for just as many years I've told them that they won't encounter functions like this outside theoretical mathematics.

But now I'm wondering whether simplified mathematical models of Euler's disk (see http://en.wikipedia.org/wiki/Euler%27s_Disk) or other idealized physical systems might involve functions in which the amplitude of some oscillatory quantity goes to zero while the frequency goes to infinity in finite time, and in particular, whether there might be "natural" examples of differentiable functions with discontinuous derivatives.

What do you think?

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