  By imposing the boundary condition associated with the boundary structure of
the null boundaries rather than the usual one, we find that the key requirement
in Harlow-Wu's algorithm fails to be met in the whole covariant phase space.
Instead, it can be satisfied in its submanifold with the null boundaries given
by the expansion free and shear free hypersurfaces in Einstein's gravity, which
can be regarded as the origin of the non-triviality of null boundaries in terms
of Wald-Zoupas's prescription. But nevertheless, by sticking to the variational
principle as our guiding principle and adapting Harlow-Wu's algorithm to the
aforementioned submanifold, we successfully reproduce the Hamiltonians obtained
previously by Wald-Zoupas' prescription, where not only are we endowed with the
expansion free and shear free null boundary as the natural stand point for the
definition of the Hamiltonian in the whole covariant phase space, but also led
naturally to the correct boundary term for such a definition.
