  In the Hamiltonian formulation, it is not a priori clear whether a symmetric
configuration will keep its symmetry during evolution. In this paper, we give
precise requirements of when this is the case and propose a symmetry
restriction to the phase space of the symmetric variables. This can often ease
computation, especially when transcending from the infinite dimensional phase
space of a field theory to a possibly finite dimensional subspace. We will
demonstrate this in the case of gravity. A prominent example is the restriction
of full Hamiltonian general relativity to the cosmological configurations of
Robertson-Walker type. We will demonstrate our procedure in this setting and
extend it to examples which appear useful in certain approaches to quantum
gravity.
