  We construct Hamiltonian dynamics of the generalized spherically symmetric
gravitational model in extended phase space. We start from the Faddeev - Popov
effective action with gauge-fixing and ghost terms, making use of gauge
conditions in differential form. It enables us to introduce missing velocities
into the Lagrangian and then construct a Hamiltonian function according a usual
rule which is applied for systems without constraints. The main feature of
Hamiltonian dynamics in extended phase space is that it can be proved to be
completely equivalent to Lagrangian dynamics derived from the effective action.
We find a BRST invariant form of the effective action by adding terms not
affecting Lagrangian equations. After all, we construct the BRST charge
according to the Noether theorem. Our algorithm differs from that by Batalin,
Fradkin and Vilkovisky, but the resulting BRST charge generates correct
transformations for all gravitational degrees of freedom including gauge ones.
Generalized spherically symmetric model imitates the full gravitational theory
much better then models with finite number of degrees of freedom, so that one
can expect appropriate results in the case of the full theory.
