Euler–Lagrange Equation

A differential equation obtained as the necessary condition for a functional defined by an integral over a domain to be stationary under compactly supported smooth variations of the dependent field.

A differential equation giving the necessary condition for a functional S[q]=∫ L(q, q̇, t) dt to be stationary under smooth variations of the generalized coordinate q with fixed endpoints; for a Lagrangian L(q,q̇,t) differentiable in q and q̇ the condition is d/dt(∂L/∂q̇) − ∂L/∂q = 0, which yields the equations of motion for finite-dimensional mechanical systems and the analogous field equations w