  A breathing mode in a Hamiltonian system is a function on the phase space
whose evolution is exactly periodic for all solutions of the equations of
motion. Such breathing modes are familiar from nonlinear dynamics in harmonic
traps or anti-de Sitter spacetimes, with applications to the physics of cold
atomic gases, general relativity and high-energy physics. We discuss the
implications of breathing modes in weakly nonlinear regimes, assuming that both
the Hamiltonian and the breathing mode are linear functions of a coupling
parameter, taken to be small. For a linear system, breathing modes dictate
resonant relations between the normal frequencies. These resonant relations
imply that arbitrarily small nonlinearities may produce large effects over long
times. The leading effects of the nonlinearities in this regime are captured by
the corresponding effective resonant system. The breathing mode of the original
system translates into an exactly conserved quantity of this effective resonant
system under simple assumptions that we explicitly specify. If the nonlinearity
in the Hamiltonian is quartic in the canonical variables, as is common in many
physically motivated cases, further consequences result from the presence of
the breathing modes, and some nontrivial explicit solutions of the effective
resonant system can be constructed. This structure explains in a uniform
fashion a series of results in the recent literature where this type of
dynamics is realized in specific Hamiltonian systems, and predicts other
situations of interest where it should emerge.
