Bialynicki-Birula and Charzynski [1] argued that the gravitational wave emitted during the merger of a black hole binary may trap particles. In this Letter we amplify their statement by describing particle motion in the wave proposed by Lukash [2] to
study anisotropic cosmological models. Bounded geodesics (found both analytically and numerically) arise when the wave is of Bianchi type VI. Their symmetries are identified.
The method proposed by Inomata and his collaborators allows us to transform a damped Caldiroli-Kanai oscillator with time-dependent frequency to one with constant frequency and no friction by redefining the time variable, obtained by solving a Ermako
v-Milne-Pinney equation. Their mapping Eisenhart-Duval lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.
Some aspects of the exotic particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models
are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.
