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The Lagrangian and Eulerian specifications of the kinematics and dynamics of the flow field are related by the material derivative (also called the Lagrangian derivative, convective derivative, substantial derivative, or particle derivative). [1] Suppose we have a flow field u, and we are also given a generic field with Eulerian specification F ...
In continuum mechanics, the material derivative [1] [2] describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum ...
The derivative of a field with respect to a fixed position in space is called the Eulerian derivative, while the derivative following a moving parcel is called the advective or material (or Lagrangian [2]) derivative. The material derivative is defined as the linear operator:
In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.The dependent variables are replaced by the value of a field at that point in spacetime (,,,) so that the equations of motion are obtained by means of an action principle, written as: =, where the action, , is a functional of the dependent ...
A Lagrangian density L (or, simply, a Lagrangian) of order r is defined as an n-form, n = dim X, on the r-order jet manifold J r Y of Y.. A Lagrangian L can be introduced as an element of the variational bicomplex of the differential graded algebra O ∗ ∞ (Y) of exterior forms on jet manifolds of Y → X.
In this reference frame, fluid parcels are labelled and followed through space and time. But also in the Eulerian frame of reference the notion of fluid parcels can be advantageous, for instance in defining the material derivative, streamlines, streaklines, and pathlines; or for determining the Stokes drift. [1]
The main points of the proof can be made clearer by considering a one-dimensional system with a Lagrangian (, ˙, ¨).The Euler–Lagrange equation is ˙ + ¨ = Non-degeneracy of means that the canonical coordinates can be expressed in terms of the derivatives of and vice versa.
In its application, the Lagrangian–Eulerian method can be accelerated using the GPUs used in common chipsets present in Nvidia and ATI Radeon graphics cards. [5] Ensuring that the moving texture always follows the velocity field of the fluid, while maintaining properties of the original texture, is key to avoid visual artifacts.