A new method for generating the canonical ensemble via continuous dynamics is presented. The new method is based on controlling the fluctuations of an arbitrary number of moments of the multidimensional Gaussian momentum distribution function. The equations of motion are non-Hamiltonian, and hence have a nonvanishing phase space compressibility. By applying the statistical mechanical theory of non-Hamiltonian systems recently introduced by the authors [M. E. Tuckerman, C. J. Mundy, and G. J. Martyna, Europhys. Lett. 45, 149 (1999)], the equations are shown to produce the correct canonical phase space distribution function. Reversible integrators for the new equations of motion are derived based on a Trotter-type factorization of the classical Liouville propagator. The new method is applied to a variety of simple one-dimensional example problems and is shown to generate ergodic trajectories and correct canonical distribution functions of both position and momentum. The new method is further shown to lead to rapid convergence in molecular dynamics based calculations of path integrals. The performance of the new method in these examples is compared to that of another canonical dynamics method, the Nose-Hoover chain method [G. J. Martyna, M. L. Klein, and M. E. Tuckerman, J. Chem. Phys. 97, 2635 (1992)]. The comparison demonstrates the improvements afforded by the new method as a molecular dynamics tool. Finally, when employed in molecular dynamics simulations of biological macromolecules, the new method is shown to provide better energy equipartitioning and temperature control and to lead to improved spatial sampling over the Nose-Hoover chain method in a realistic application.