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The 24 pages that link to Euler–Lagrange Equation, each with the reason it gives.

• Leonhard EulerBroader topic: Euler derived a general condition that became the calculus of variations’ main tool.

• Lagrangian mechanicsBroader topic: Applying it to the action yields the equations for generalized coordinates.

• Noether's theoremRelated: The theorem combines these equations with symmetry to derive a conserved current.

• Joseph-Louis LagrangeBroader topic: It expresses the central differential condition behind Lagrange’s variational method.

• Calculus of variationsBroader topic: It supplies the central necessary condition for unconstrained extrema.

• Principle of least actionBroader topic: Applying it to the action yields the equations of motion.

• Fermat's principleRelated: Applying this variational equation to optical travel time yields ray paths.

• Christoffel symbolsRelated: Geodesic equations with Christoffel terms follow from a metric-based variational principle.

• Hamilton–Jacobi equationCompared with: It determines paths by varying trajectories; Hamilton–Jacobi theory solves for the action across space and time.

• Action (physics)Related: Applied to the action, it yields the equations of motion.

• Hamilton's principleRelated: Varying the action produces this equation for the system's motion.

• Isoperimetric problemRelated: A multiplier converts the constrained problem into an ordinary variational problem governed by this equation.

• Classical field theoryRelated: Its field form produces equations of motion from a Lagrangian density.

• Direct method in the calculus of variationsCompared with: It characterizes candidates, while the direct method proves a minimizer exists without first solving that equation.

• Pierre Louis MaupertuisRelated: It expresses the general variational framework later used to formalize least-action mechanics.

• Analytical mechanicsRelated: It converts the stationary-action condition into local equations of motion.

• Noether's second theoremRelated: The theorem’s identities constrain these equations when the action has arbitrary-function symmetries.

• Leibniz integral ruleRelated: Deriving it differentiates an integral functional with respect to a perturbation parameter.

• Scalar field theoryRelated: Varying the scalar-field action yields its equation of motion.

• Brachistochrone curveRelated: Applying it to travel time yields the differential equation for the optimal path.

• Clairaut's relationRelated: Applying it to the geodesic energy on a rotationally symmetric metric produces the invariant.

• Fundamental lemma of the calculus of variationsRelated: The lemma turns the first variation's vanishing for all perturbations into this equation.

• Pontryagin's maximum principleCompared with: The maximum principle generalizes this variational condition to controlled dynamical systems.

• Schwinger's quantum action principleRelated: The principle yields quantum equations of motion related to this classical variational result.