  We explore the manipulation in phase space of many-body wavefunctions that
exhibit self-similar dynamics, under the application of sudden force and/or in
the presence of a constant acceleration field. For this purpose, we work out a
common theoretical framework based on the Wigner function. We discuss squeezing
in position space, phase space rotation and its implications in cooling for
both non-interacting and interacting gases, and time reversal operation. We
discuss various optical analogies and calculate the role of spherical-like
aberration in cooling protocols. We also present the equivalent of a spin-echo
technique to improve the robustness of velocity dispersion reduction protocols.
