Using the standard canonical formalism, the equations of mechanics and
kinetics in the Friedmann-Lemaitre-Robertson-Walker (FLRW) space-times in
Cartesian coordinates have been obtained. The transformation law of the
generalized momentum under the shift of the origin of the coordinate system has
been found, and the form invariance of the Hamiltonian function relative to the
shift transformation has been proved. The general solution of the collisionless
Boltzmann equation has been found. In the case of the homogeneous distribution
the solutions of the kinetic equation for several simple, but important for
applications, cases have been obtained
