HAMILTONIAN FLUID MECHANICS

where x = φ(X, T ) is the position of the material point X at time T . We will refer to X as a material, or Lagrangian point, and x as a spatial or Eulerian point. If Ω ⊂ B is a material region, we denote by ΩT = φ (Ω, T ) the spatial region occupied by the material points in Ω at time T . We assume that the deformation φ is smooth, with a smooth inverse X = Φ(x, t). Here, we write T = t for the time; the partial derivative ∂T will denote the material time-derivative, taken holding X fixed, while ∂t will denote the spatial timederivative, taken holding x fixed. Abusing notation, we will usually denote the deformation and its inverse by x(X, T ) and X(x, t), respectively. We denote the Cartesian coordinates of the position vector of a point, chosen with respect to some convenient origin, in the spatial and reference configurations by x = ( x, . . . , x, . . . , x ) , X = ( X, . . . , X, . . . , X ) ,