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-Wus 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 Einsteins gravity, which can be regarded as the origin of the non-triviality of null boundaries in terms of Wald-Zoupass prescription. But nevertheless, by sticking to the variational principle as our guiding principle and adapting Harlow-Wus 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.
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