  Particles interacting with a prescribed quasimonochromatic gravitational wave
(GW) exhibit secular (average) nonlinear dynamics that can be described by
Hamilton's equations. We derive the Hamiltonian of this "ponderomotive"
dynamics to the second order in the GW amplitude for a general background
metric. For the special case of vacuum GWs, we show that our Hamiltonian is
equivalent to that of a free particle in an effective metric, which we
calculate explicitly. We also show that already a linear plane GW pulse
displaces a particle from its unperturbed trajectory by a finite distance that
is independent of the GW phase and proportional to the integral of the pulse
intensity. We calculate the particle displacement analytically and show that
our result is in agreement with numerical simulations. We also show how the
Hamiltonian of the nonlinear averaged dynamics naturally leads to the concept
of the linear gravitational susceptibility of a particle gas with an arbitrary
phase-space distribution. We calculate this susceptibility explicitly to apply
it, in a follow-up paper, toward studying self-consistent GWs in inhomogeneous
media within the geometrical-optics approximation.
