  We study two-dimensional systems with Galilean invariance gapped under
magnetic fields. When such quantum Hall systems are coupled with external
sources for charge, energy, and momentum currents, they exhibit invariance
under the Milne boost as well as under the gauge and general coordinate
transformations. We construct the most general effective action consistent with
all the symmetries in the derivative expansion, where an electric field is
regarded as order of unity so as to allow for nonlinear responses. The
resulting action is shown to consist of four terms proportional to the Hall
conductivity and viscosity and the energy density and magnetization. We then
compute the local currents induced by electromagnetic fields, revealing
universal relations among distinct kinds of responses. In particular, we find
the Hall conductivity determining the longitudinal conductivity at nonzero
frequency and the Hall viscosity contributing to the nonlinear electrothermal
conductivity at nonzero wave number.
