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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:
Derivation of the Lagrangian and Eulerian finite strain tensors A measure of deformation is the difference between the squares of the differential line element d X {\displaystyle d\mathbf {X} \,\!} , in the undeformed configuration, and d x {\displaystyle d\mathbf {x} \,\!} , in the deformed configuration (Figure 2).
The Euler equations can be formulated in a "convective form" (also called the "Lagrangian form") or a "conservation form" (also called the "Eulerian form"). The convective form emphasizes changes to the state in a frame of reference moving with the fluid.
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 ...
The Euler–Lagrange equation was developed in connection with their studies of the tautochrone problem. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in ...
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 ...