The properties of the Mathieu equation are reviewed in order to discus
s some of the applications that have appeared in recent years. Those m
entioned are: vibrations in an elliptic drum, the inverted pendulum, t
he radio frequency quadrupole, frequency modulation, stability of a fl
oating body, alternating gradient focusing, the Paul trap for charged
particles, and the mirror trap for neutral particles. (C) 1996 America
n Association of Physics Teachers.