We present methods to compute thermodynamic properties of classical systems which involve extending the phase space by two degrees of freedom. These two additional degrees of freedom are used to replicate the coupling of the original system to the infinite degrees of freedom of a heat bath. In the extended phase space, the trajectories are ergodic. This feature enables one to replace phase space averages by time averages, which are extremely simple to compute. We examine phase space patterns, thermal distributions, correlations, ergodicity, Lyapunov exponents, mixing, and rate of convergence in an analysis of several simple systems. © 1990.