  Recently a {\it local} true (completely gauge fixed) Hamiltonian for
spherically symmetric collapse was derived in terms of Ashtekar variables. We
show that such a local Hamiltonian follows directly from the geometrodynamics
of gravity theories that obey a Birkhoff theorem and possess a mass function
that is constant on the constraint surface in vacuum. In addition to clarifying
the geometrical content, our approach has the advantage that it can be directly
applied to a large class of spherically symmetric and 2D gravity theories,
including $p$-th order Lovelock gravity in D dimensions. The resulting
expression for the true local Hamiltonian is universal and remarkably simple in
form.
