  We apply the Dirac procedure for constrained systems to the
Arnowitt-Deser-Misner formalism linearized around the Friedmann-Lemaitre
universe. We explain and employ some basic concepts such as Dirac observables,
Dirac brackets, gauge-fixing conditions, reduced phase space, physical
Hamiltonian and physical dynamics. In particular, we elaborate on the key
concept which is the canonical isomorphism between different gauge-fixing
surfaces. We apply our formalism to describe the reduced phase space of
cosmological perturbations in some popular in the literature gauges. Our
formalism is first developed for the universe with a single fluid and then
extended to the multi-fluid case. The obtained results are a starting point for
complete quantization of the cosmological perturbations and the cosmological
background. Our approach may be used in future to derive the reduced phase
space of higher order perturbations and in more generic cosmological
spacetimes.
