Euclidean Quantum Gravity from Variational Dynamics

A variational phase space is constructed for a compact and piecewise flat Riemannian manifold. An extended action functional is provided such that the variational dynamics generate a symplectic flow on the phase space. This symplectic flow is numerically integrated as it evolves with respect to the variational parameter. Assuming ergodicity, the resulting flow samples the Euclidean path integral.