text: The Lagrangian motion of a fluid parcel with position vector x = ξ(α, t) in the Eulerian coordinates is given by
  where
  : ∂ξ/∂t is the partial derivative of ξ(α, t) with respect to t,
  : ξ(α, t) is the Lagrangian position vector of a fluid parcel,
  : u(x, t) is the Eulerian velocity,
  : x is the position vector in the Eulerian coordinate system,
  : α is the position vector in the Lagrangian coordinate system,
  : t is time.
  Often, the Lagrangian coordinates α are chosen to coincide with the Eulerian coordinates x at the initial time t = t:
  If the average value of a quantity is denoted by an overbar, then the average Eulerian velocity vector ū and average Lagrangian velocity vector ū are
  Different definitions of the average may be used, depending on the subject of study (see ergodic theory):
  * time average,
  * space average,
  * ensemble average,
  * phase average.
  The Stokes drift velocity ū is defined as the difference between the average Eulerian velocity and the average Lagrangian velocity:
  In many situations, the mapping of average quantities from some Eulerian position x to a corresponding Lagrangian position α forms a problem. Since a fluid parcel with label α traverses along a path of many different Eulerian positions x, it is not possible to assign α to a unique x.
  A mathematically sound basis for an unambiguous mapping between average Lagrangian and Eulerian quantities is provided by the theory of the generalized Lagrangian mean (GLM) by Andrews and McIntyre (1978).