While Eulerian description investigates the physics encoded in an observable O(t, [??]) at a fixed spatial position [??], position is not an independent variable but instead [mathematical expression not reproducible] in the Lagrangian description.
In order to employ the Poisson bracket to derive the fluid dynamics and furthermore to incorporate nontrivial bracket algebra, starting from the single particle picture is necessary to find the correct and convenient map from Lagrangian to Eulerian description. Our formalism for this map is based on the dynamics of single particle and always starts from a system with finite number of particles and then derives the relevant expressions and canonical algebras in the limit of infinite number of particles.
The continuum mechanics usually adopt two classical descriptions of motion: the Lagrangian description and the Eulerian description
(see e.g., Malvern ).
In a relativistic aether, using the Levi-Civita connection produces the Lagrangian description while using the Weitzenbock connection produces the Eulerian description
The canopy (Lagrangian description) and fluid domain (Eulerian description
) interpenetrate with each other.
Because of the nature of filling process, Eulerian description is used in modeling of the mold filling.
Under the frame of Eulerian description, the filling state is defined element by element for the integration of different behaviors.
Fluid mechanics, on the other hand, tends to rely on a Eulerian description
of motion, where the motion of fluids is characterized as a function of time.
Our primary aim is to develop a new foundational world-geometry based on the intuitive notion of a novel, fully naturalized kind of Finsler geometry, which extensively mimics the Eulerian description
of the mechanics of continuous media with special emphasis on the world-velocity field, in the sense that the whole space-time continuum itself is taken to be globally dynamic on both microscopic and macroscopic scales.
Coverage encompasses Lagrangian and Eulerian descriptions
of hydrodynamic systems, spaces and operators, Lagrangian-Eulerian existence theorems, and critical dissipative active scalars.