  The Weiss variational principle in mechanics and classical field theory is a
variational principle which allows displacements of the boundary. We review the
Weiss variation in mechanics and classical field theory, and present a novel
geometric derivation of the Weiss variation for the gravitational action: the
Einstein-Hilbert action plus the Gibbons-Hawking-York boundary term. In
particular, we use the first and second variation of area formulas (we present
a derivation accessible to physicists in an appendix) to interpret and vary the
Gibbons-Hawking-York boundary term. The Weiss variation for the gravitational
action is in principle known to the Relativity community, but the variation of
area approach formalizes the derivation, and facilitates the discussion of time
evolution in General Relativity. A potentially useful feature of the formalism
presented in this article is that it avoids an explicit 3+1 decomposition in
the bulk spacetime.
