Euler-Lagrange equation

Named after the Swiss mathematician and physicist Leonhard Euler (1707–1783), and the Italian-born French mathematician and astronomer Joseph Louis Lagrange (1736–1813).

Euler-Lagrange equation (plural Euler-Lagrange equations)

1. (mechanics, analytical mechanics) A differential equation which describes a function ${\displaystyle \mathbf {q} (t)}$ which describes a stationary point of a functional, ${\displaystyle S(\mathbf {q} )=\int L(t,\mathbf {q} (t),\mathbf {\dot {q}} (t))\,dt}$, which represents the action of ${\displaystyle \mathbf {q} (t)}$, with ${\displaystyle L}$ representing the Lagrangian. The said equation (found through the calculus of variations) is ${\displaystyle {\partial L \over \partial \mathbf {q} }={d \over dt}{\partial L \over \partial \mathbf {\dot {q}} }}$ and its solution for ${\displaystyle \mathbf {q} (t)}$ represents the trajectory of a particle or object, and such trajectory should satisfy the principle of least action.