Collection: Euler–Lagrange Equations

In the calculus of variations and classical mechanics, the Euler–Lagrange equations is a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.

Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero.

This equation is used to analyze everything from the shape of a soap bubble to the trajectory of a rocket around a black hole. More than just an equation, it has been said that it’s actually a recipe to generate an infinite variety of possible physical laws.

Despite its many applications, the equation is deceptively short and simple. The way that all classical physics can be expressed and understood within a single framework like that helps reveal deep links between seemingly different phenomena.