Euler–Lagrange equation

E54267

differential equation equation in the calculus of variations necessary condition for an extremum of a functional

The Euler–Lagrange equation is a fundamental differential equation in the calculus of variations that provides the condition for a function to make a functional stationary, forming the basis of Lagrangian mechanics and many physical theories.

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All labels observed (6)

Label	Occurrences
Euler–Lagrange equations	7
Euler–Lagrange equation canonical	5
Euler equations	1
Euler–Lagrange differential equation	1
Euler–Lagrange equation for fields	1
Euler–Lagrange equation with constraints	1

How this entity was disambiguated

This entity first appeared as the object of triple T426762 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Euler–Lagrange equation
Context triple: [Leonhard Euler, notableWork, Euler–Lagrange equation]

A. Euler equations
The Euler equations are fundamental partial differential equations in fluid dynamics that describe the motion of an ideal (inviscid) fluid without viscosity.
B. Noether's theorem
Noether's theorem is a fundamental result in theoretical physics and mathematics that links continuous symmetries of a physical system to corresponding conservation laws, such as energy or momentum conservation.
C. Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
D. Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
E. Hessian forces
Hessian forces were German auxiliary troops hired by the British Crown during the American Revolutionary War, known for their disciplined fighting and prominent role in key battles such as Trenton.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Euler–Lagrange equation
Target entity description: The Euler–Lagrange equation is a fundamental differential equation in the calculus of variations that provides the condition for a function to make a functional stationary, forming the basis of Lagrangian mechanics and many physical theories.

A. Euler equations
The Euler equations are fundamental partial differential equations in fluid dynamics that describe the motion of an ideal (inviscid) fluid without viscosity.
B. Noether's theorem
Noether's theorem is a fundamental result in theoretical physics and mathematics that links continuous symmetries of a physical system to corresponding conservation laws, such as energy or momentum conservation.
C. Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
D. Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
E. Hessian forces
Hessian forces were German auxiliary troops hired by the British Crown during the American Revolutionary War, known for their disciplined fighting and prominent role in key battles such as Trenton.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	differential equation ⓘ equation in the calculus of variations ⓘ necessary condition for an extremum of a functional ⓘ
appliesTo	functionals of the form ∫ F(x,y,y′) dx ⓘ problems with fixed boundary conditions ⓘ problems with natural boundary conditions ⓘ
assumes	sufficient smoothness of the admissible functions ⓘ
basedOn	principle of stationary action ⓘ
canBe	a system of coupled differential equations ⓘ
captures	stationarity of the action functional ⓘ
field	Lagrangian mechanics ⓘ calculus of variations ⓘ classical mechanics ⓘ mathematical physics ⓘ
fieldTheoreticForm	∂ℒ/∂φ − ∂_μ(∂ℒ/∂(∂_μφ)) = 0 ⓘ
generalForm	∂L/∂qᵢ − d/dt(∂L/∂q̇ᵢ) = 0 ⓘ
generalizes	Fermat’s principle in optics ⓘ linked to: Fermat’s principle of least time geodesic equation as shortest path condition ⓘ
gives	condition for a functional to be stationary ⓘ
hasVariant	Euler–Lagrange equation for fields ⓘ linked to: Euler–Lagrange equation Euler–Lagrange equation with constraints ⓘ linked to: Euler–Lagrange equation
historicalDevelopment	formulated in the 18th century ⓘ
involves	Lagrangian function ⓘ
is	a necessary condition for extrema but not generally sufficient ⓘ
mathematicalDomain	analysis ⓘ differential equations ⓘ
namedAfter	Joseph-Louis Lagrange ⓘ Leonhard Euler ⓘ
relatedTo	Hamiltonian mechanics ⓘ Hamilton–Jacobi equation ⓘ Hamilton’s principle ⓘ linked to: principle of least action Lagrangian mechanics ⓘ Noether’s theorem ⓘ linked to: Noether's theorem
requires	differentiability of the integrand with respect to its arguments ⓘ
standardForm	∂F/∂y − d/dx(∂F/∂y′) = 0 ⓘ
typeOf	second-order ordinary differential equation in mechanics ⓘ
usedFor	deriving equations of motion in Lagrangian mechanics ⓘ deriving field equations in classical field theory ⓘ finding stationary points of functionals ⓘ optimization problems with integral cost functionals ⓘ
usedIn	classical field theory ⓘ electromagnetism ⓘ engineering optimization ⓘ general relativity ⓘ geodesic problems in differential geometry ⓘ optimal control theory ⓘ quantum field theory ⓘ

How these facts were elicited