In this work, we employ a Hamiltonian-based procedure to derive a generalized Nose-Hoover thermostat, which generates rigorous trajectories, corresponding to the canonical ensemble, in the absence or presence of external forces. Specifically, we prove that rigorous trajectories are generated regardless of whether the total linear momentum of the system is zero, constant, or time-dependent. Through the use of peculiar and center-of-mass coordinates, we develop a thermostat that is consistent with the definition of temperature through the equipartition theorem. The generalized algorithm reduces to the conventional Nose-Hoover thermostat under the constraints that (i) the external forces are absent and (ii) the total linear momentum is zero. We show that the generalized algorithm satisfies the two criteria for rigor (Hamiltonian and non-Hamiltonian) that exist in the literature. Finally, we provide some numerical examples demonstrating the success of the generalized algorithm. (C) 2007 Elsevier B.V. All rights reserved.