The n-variable Fokker-Planck equation can be written in an equivalent form as a system of n + 1 first-order equations by introducing as auxiliary variables the components of the drift velocity R,. The stationary state defines a stationary R uniquely, which allows an intrinsic classification of the stationary states in terms of the properties of R, without reference to detailed balance. This representation is very appropriate for the study of questions such as the existence of stationary states and their small and large noise asymptotics, as well as for the construction of models having some specified behavior. R provides also a classification of the dynamics, which corresponds to the hermiticity properties of the associated eigenvalue problem.