  Starting from a local action for mimetic gravity that includes higher
derivatives of a scalar field $\phi$, we derive a gauge-fixed canonical action
of the theory in the ADM canonical formalism in the time gauge $\phi=t$. This
reduced action reveals (i) a non-vanishing conserved physical Hamiltonian that
is a sum of two terms, the expression for the Hamiltonian constraint of general
relativity and a function of the expansion scalar, and (ii) a reduced
symplectic structure that geometrically provides the Dirac brackets. As
applications of our general analysis, we compute the physical Hamiltonians and
canonical equations for perturbations around Minkowski spacetime, homogeneous
cosmologies, and spherically symmetric spacetimes.
