  Theoretical studies of nonequilibrium systems are complicated by the lack of
a general framework. In this work we first show that a transformation
introduced by Ao recently (J. Phys. A {\bf 37}, L25 (2004)) is related to
previous works of Graham (Z. Physik B {\bf 26}, 397 (1977)) and Eyink {\it et
al.} (J. Stat. Phys. {\bf 83}, 385 (1996)), which can also be viewed as the
generalized application of the Helmholtz theorem in vector calculus. We then
show that systems described by ordinary stochastic differential equations with
white noise can be mapped to thermostated Hamiltonian systems. A steady-state
of a dissipative system corresponds to the equilibrium state of the
corresponding Hamiltonian system. These results provides a solid theoretical
ground for corresponding studies on nonequilibrium dynamics, especially on
nonequilibrium steady state. The mapping permits the application of established
techniques and results for Hamiltonian systems to dissipative non-Hamiltonian
systems, those for thermodynamic equilibrium states to nonequilibrium steady
states. We discuss several implications of the present work.
