Euler–Lagrange equation

English

Wikipedia

Etymology

Named after Leonhard Euler (1707–1783), Swiss mathematician and physicist, and Joseph Louis Lagrange (1736–1813), French mathematician and astronomer — originally from Italy.

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.