Euler-Lagrange equation

English

Etymology

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

Noun

Euler-Lagrange equation (plural Euler-Lagrange equations)

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