In the Lagrangian approach to fluid flow, one particle is chosen and is followed as it moves through space with time. The line traced out by that one particle is called a particle pathline.

Joseph Lagrange was an Italian-French mathematician who contributed to the theory of numbers and to analytic and celestial mechanics. He was born in Turin, Italy, in January 1736. His family was well off but lost much of its wealth through unwise investments.

Lagrange largely taught himself mathematics, which he became interested in after reading a memoir by the English astronomer Edmond Halley (for whom Halley's comet is named) that discussed the use of algebra in optics. He began teaching mathematics while still a teenager, and helped found the Turin Academy of Sciences.

By the early 1760s, he was known as one of the greatest living mathematicians. In 1764 he received a prize from the Paris Academy of Sciences for an essay he wrote about the apparent oscillation of the moon (called "libration") that makes its features seem to appear and disappear.

In 1766, Leonhard Euler and Jean d'Alembert, other highly respected mathematicians, recommended Lagrange for a post at Frederick the Great's Berlin Academy. Lagrange stayed in Berlin until 1787. While there he published a large number of papers on topics including astronomy, the stability of the solar system, mechanics, dynamics, fluid mechanics, probability, and the foundations of the calculus. His paper "Réflexions sur la résolution algébrique des équations" ("Reflections on the Algebraic Resolution of Equations") was the first to consider the roots of an equation as abstract quantities rather than as numerical values.

When Frederick died, Lagrange moved to Paris at the invitation of Louis XVI, where he joined the Académie des Sciences in Paris. There he managed to stay aloof from the political turmoil that soon overwhelmed the country. While in Paris, he contributed to the theory of fluid flow, introducing the Lagrangian function important for aerodynamics, in his work Mécanique analytique. This treatise synthesized the research that had been done in mechanics since the theories of Sir Isaac Newton. Lagrange presented the law of virtual work, and from that one principle, by the aid of the calculus of variations, deduced both solid and fluid mechanics.

In 1790, Lagrange joined a committee working on establishing the metric system and he advocated use of a decimal base. In 1797, he became a professor of mathematics at the new École Polytechnique. His lectures were published as the first textbooks on real analytic functions. Although other academicians had been expelled or executed, Lagrange was specifically exempted from the decree that all foreign-born individuals leave France at the urging of the famous French chemist Antoine Lavoisier (who himself was executed).

Lagrange was honored by Napoleon Bonaparte, who named him to the Legion of Honour and a count of the empire in 1808. On April 3, 1813, he was named grand croix of the Ordre Impérial de la Réunion. He died a week later.