  A general formalism is developed for constructing modified Hamiltonian
dynamical systems which preserve a canonical equilibrium distribution by adding
a time evolution equation for a single additional thermostat variable. When
such systems are ergodic, canonical ensemble averages can be computed as
dynamical time averages over a single trajectory. Systems of this type were
unknown until their recent discovery by Hoover and colleagues. The present
formalism should facilitate the discovery, construction, and classification of
other such systems by encompassing a wide class of them within a single unified
framework. This formalism includes both canonical and generalized Hamiltonian
systems in a state space of arbitrary dimensionality (either even or odd), and
therefore encompasses both few- and many-particle systems. Particular attention
is devoted to the physical motivation and interpretation of the formalism,
which largely determine its structure. An analogy to stochastic thermostats and
fluctuation-dissipation theorems is briefly discussed.
