In an expanding universe, what doesn't expand?

The expansion of the universe is often viewed as a uniform stretching of space that would affect compact objects such as atoms and stars, as well as the separation of galaxies. One usually hears that bound systems do not take part in the general expansion, but a much more subtle question is whether bound systems expand partially. In this paper, a definitive answer is given for a very simple system: a classical "atom" bound by electrical attraction. With a mathematical description appropriate for undergraduate physics majors, we show that this bound system either completely follows the cosmological expansion, or, after initial transients, completely ignores it. This all-or-nothing behavior can be understood using analysis techniques used in junior-level mechanics. We also demonstrate that this simple description is a justifiable approximation of the relativistically correct formulation of the problem.