  Neutrino-neutrino interactions can lead to collective flavour conversion
effects in supernovae and in the early universe. We demonstrate that the case
of "bipolar" oscillations, where a dense gas of neutrinos and antineutrinos in
equal numbers completely converts from one flavour to another even if the
mixing angle is small, is equivalent to a pendulum in flavour space. Bipolar
flavour conversion corresponds to the swinging of the pendulum, which begins in
an unstable upright position (the initial flavour), and passes through
momentarily the vertically downward position (the other flavour) in the course
of its motion. The time scale to complete one cycle of oscillation depends
logarithmically on the vacuum mixing angle. Likewise, the presence of an
ordinary medium can be shown analytically to contribute to a logarithmic
increase in the bipolar conversion period. We further find that a more complex
(and realistic) system of unequal numbers of neutrinos and antineutrinos is
analogous to a spinning top subject to a torque. This analogy easily explains
how such a system can oscillate in both the bipolar and the synchronised mode,
depending on the neutrino density and the size of the neutrino-antineutrino
asymmetry. Our simple model applies strictly only to isotropic neutrino gasses.
In more general cases, and especially for neutrinos streaming from a supernova
core, different modes couple to each other with unequal strength, an effect
that can lead to kinematical decoherence in flavour space rather than
collective oscillations. The exact circumstances under which collective
oscillations occur in non-isotropic media remain to be understood.
