  For a given timelike displacement vector the covariant Hamiltonian
quasi-local energy expression requires a proper choice of reference spacetime.
We propose a program for determining the reference by embedding a neighborhood
of the two-sphere boundary in the dynamic spacetime into a Minkowski reference,
so that the two sphere is embedded isometrically, and then extremizing the
energy to determine the embedding variables. Applying this idea to
Schwarzschild spacetime, we found that for each given future timelike
displacement vector our program gives a unique energy value. The static
observer measures the maximal energy. Applied to the
Friedmann-Lemaitre-Robertson-Walker spacetime, we find that the maximum energy
value is nonnegative; the associated displacement vector is the unit dual mean
curvature vector, and the expansion of the two-sphere boundary matches that of
its reference image. For these spherically symmetric cases the reference
determined by our program is equivalent to isometrically matching the geometry
at the two-sphere boundary and taking the displacement vector to be orthogonal
to the spacelike constant coordinate time hypersurface, like the timelike
Killing vector of the Minkowski reference.
