EulerLagrangeGleichung

The Euler-Lagrange equation, known in German as the Euler-Lagrange-Gleichung, is the condition that a trajectory extremizes the action in Lagrangian mechanics. It follows from Hamilton's principle, with action S = ∫ L(q_i, q̇_i, t) dt, where L is the Lagrangian function of generalized coordinates q_i, their velocities q̇_i, and time.

For a system described by generalized coordinates q_i (i = 1, …, n), the equation of motion is d/dt (∂L/∂q̇_i) − ∂L/∂q_i = 0 for each i.

In field theory, where fields φ(x, t) replace coordinates, the Euler-Lagrange equations generalize to ∂μ (∂L/∂(∂μ φ)) − ∂L/∂φ = 0, with μ running over spacetime indices. This yields equations such as the Klein-Gordon, Maxwell, and Einstein field equations in appropriate theories.

Derivation: vary the action with respect to q_i(t), set δS = 0, integrate by parts, and assume fixed endpoints in time; this yields the Euler-Lagrange equations. The formalism is equivalent to Newton’s laws under suitable conditions and forms the basis of Hamiltonian mechanics via the Legendre transform.

Uses: deriving equations of motion in classical mechanics, electromagnetism, general relativity, and modern theoretical physics; also used in optimization problems and analytical mechanics.