Euler-Lagrange equation

The Euler-Lagrange equation is a fundamental concept in the calculus of variations, a branch of mathematics focused on finding functions that optimize (maximize or minimize) the value of integrals. This field emerged from problems like the isoperimetric problem and the brachistochrone problem, which seek shapes that enclose the most area for a given perimeter or take the least time to traverse, respectively.12

Leonhard Euler developed a differential equation, later known as the Euler-Lagrange equation, which is crucial for determining minimizing arcs between two points. This equation is a cornerstone for expressing scientific laws as variational principles, where certain integrals are minimized or maximized. It plays a significant role in fields like mechanics, elasticity, and fluid dynamics, notably in the formulation of the Navier-Stokes equation.23