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 general relativity in its Hamiltonian formulation to the cosmological configurations of Robertson–Walker type. We will demonstrate our procedure in this setting and extend it to examples that appear useful in certain approaches to quantum gravity.