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.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Euler–Lagrange equation canonical | 4 |
| Euler–Lagrange equations | 4 |
| 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 of least time
ⓘ
surface form:
Fermat’s principle in optics
geodesic equation as shortest path condition ⓘ |
| gives | condition for a functional to be stationary ⓘ |
| hasVariant |
Euler–Lagrange equation
self-linksurface differs
ⓘ
surface form:
Euler–Lagrange equation for fields
Euler–Lagrange equation self-linksurface differs ⓘ
surface form:
Euler–Lagrange equation with constraints
|
| 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 ⓘ principle of least action ⓘ
surface form:
Hamilton’s principle
Lagrangian mechanics ⓘ Noether's theorem ⓘ
surface form:
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Euler–Lagrange equation Description of subject: 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.
Referenced by (12)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Euler–Lagrange equations
this entity surface form:
Euler–Lagrange equation with constraints
this entity surface form:
Euler–Lagrange equation for fields
this entity surface form:
Euler–Lagrange equations
subject surface form:
Invariante Variationsprobleme
this entity surface form:
Euler–Lagrange equations
this entity surface form:
Euler equations
this entity surface form:
Euler–Lagrange equations
this entity surface form:
Euler–Lagrange differential equation