 Lagrangian coordinate

A Lagrangian coordinate labels a material particle and follows its identity as the body moves. Taking the particle's position $\mathbf{X}$ in a reference configuration as its label gives the deformation map $\mathbf{x}=\mathbf{\chi }(\mathbf{X},t)$. Differentiation at fixed $\mathbf{X}$ gives that particle's velocity and acceleration.

A floating marker in water approximates a Lagrangian trajectory, while a fixed current meter records an Eulerian coordinate description. The distinction is useful because a particle can cross regions with different velocity even when every fixed-location reading is steady. For a conserved material element, mass conservation becomes $\rho J={\rho }_0$, relating its current mass density to its changing volume. Material labels are especially natural for elasticity, where the same region's initial and final lengths are compared to define strain.