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).