Neutrinos in dense environments undergo collective pair conversions νeν¯e↔νxν¯x, where x is a nonelectron flavor, due to forward scattering off each other that may be a crucial ingredient for supernova explosions. Depending on the flavor-dependent local angular distributions of the neutrino fluxes, the conversion rate can be “fast,” i.e., of the order μ=2GFnν, which can far exceed the usual neutrino oscillation frequency ω=Δm2/(2E). Until now, this surprising nonlinear phenomenon has only been understood in the linear regime and explored further using numerical experiments. We present an analytical treatment of the simplest system that exhibits fast conversions, and show that the conversion can be understood as the dynamics of a particle rolling down in a quartic potential, governed dominantly by μ but seeded by slower subleading effects.