Euler–Lagrange Equation

The Euler–Lagrange equation is a differential equation satisfied by stationary points of a functional, relating a function to its derivatives through the functional’s integrand.

Calculus of Variations: The mathematical study of extrema and stationary points of functionals. The equation is a central tool for finding stationary functions in this broader subject.
Functional: A mapping that assigns a scalar, or another specified value, to a function. The equation concerns stationary values of functionals rather than ordinary functions of numbers.
Principle of Least Action: A physical principle stating that classical trajectories make the action stationary under allowed variations. Applying the equation to the action produces the equations of motion of classical systems.
Fermat's Principle: The optical principle that light follows paths making travel time stationary. It is a physical variational principle, while the Euler–Lagrange equation is the condition derived from such principles.
Functional Derivative: A derivative that measures how a functional changes under an infinitesimal change in its input function. Setting the functional derivative to zero yields the stationarity condition encoded by the equation.
Lagrangian (field theory): A function of fields and their derivatives whose integral defines an action. In field theory, this density supplies the integrand from which the Euler–Lagrange equations are derived.
Euler–Lagrange Equation for Fields: The field-theoretic form of the Euler–Lagrange equation, involving derivatives with respect to spacetime coordinates. It extends the equation from paths to fields and underlies classical field equations.
Lagrange Multipliers: A method for finding extrema of functions subject to equality constraints. It handles finite-dimensional constrained extrema, unlike the differential condition for function-valued inputs.
First Variation: The first-order change in a functional caused by a small perturbation of its input. The equation follows by requiring this first-order change to vanish for every admissible perturbation.
Partial Derivative: A derivative with respect to one variable while other variables are held fixed. The equation differentiates the integrand with respect to the function and its derivative separately.

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