Euler-Lagrange equation

noun

UK /ˈɔɪlə ləˈɡrɑːnʒ ɪˈkweɪʒən/

US /ˈɔɪlər ləˈɡrɑːnʒ ɪˈkweɪʒən/

A math rule that helps find the best path or shape, like the quickest way down a hill.

The Euler-Lagrange equation tells us the curve a bead follows when it slides down a wire in the shortest time.

SYNONYMS

variational equation Euler equation Lagrange equation stationarity condition

RELATED WORDS

calculus of variations Lagrangian action principle functional derivative Hamilton's principle

OTHER MEANINGS

2. The differential equation arising from making a functional stationary, written as ∂L/∂q − d/dt(∂L/∂q̇) = 0.

Solving the Euler-Lagrange equation for the Lagrangian gives the equations of motion.

3. In field theory, the partial differential equation that extremizes an action depending on fields and their derivatives.

The Euler-Lagrange equation for the electromagnetic field yields Maxwell's equations.

4. A general stationarity condition used in optimization and mechanics to find extrema of functionals.

Setting the first variation to zero produces the Euler-Lagrange equation.