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.

Observed surface forms (3)

Surface form	As subject	As object
Euler–Lagrange equation for fields	0	1
Euler–Lagrange equation with constraints	0	1
Euler–Lagrange equations	0	1

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 →

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Noether's theorem → basedOn → Euler–Lagrange equation →

this entity surface form: Euler–Lagrange equations

Euler–Lagrange equation → hasVariant → Euler–Lagrange equation self-linksurface differs →

this entity surface form: Euler–Lagrange equation with constraints

Euler–Lagrange equation → hasVariant → Euler–Lagrange equation self-linksurface differs →

this entity surface form: Euler–Lagrange equation for fields

Leonhard Euler → notableWork → Euler–Lagrange equation →