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Balance Equations: Eulerian and Lagrangian Formulations

The term Lagrangian Formulation is usually used to describe a balance equation written as a function of the coordinates in the reference coordinate system, while the term Eulerian Formulation is usually used to describe an equation written as a function of the coordinates in the deformed coordinate system. In a Lagrangian formulation, the coordinates of a vector X \in \sigma_0 \subset \mathbb{R}^3 correspond to a material point and are fixed in time, so differential equations written in terms of X describe the material rates of change of the different variables appearing in the differential equation. In an Eulerian formulation, quantities are written as a function of the spatial coordinates of the vector x \in \sigma \subset \mathbb{R}^3, thus, the quantities appearing in the differential equation point to a position in space rather than a material point. Traditionally, equations for solids usually use the Lagrangian formulations while those for fluids use the Eulerian formulations.

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