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Measure of the light propagating through an optical system
In optics the Lagrange invariant is a measure of the light propagating through an optical system. It is defined by H = n u ¯ y − n u y ¯ {\displaystyle
Lagrange_invariant
bound Lagrange form Lagrange form of the remainder Lagrange interpolation Lagrange invariant Lagrange inversion theorem Lagrange multiplier Augmented
List of things named after Joseph-Louis Lagrange
List_of_things_named_after_Joseph-Louis_Lagrange
Measure of the "spread" of light in an optical system
etendue as the source. The etendue is related to the Lagrange invariant and the optical invariant, which also share the property of being constant in an
Etendue
Smith–Helmholtz invariant is closely related to the Lagrange invariant and the optical invariant. The Smith–Helmholtz is the optical invariant restricted to
Smith–Helmholtz_invariant
Formulation of classical mechanics
t)\end{aligned}}} and by the chain rule for partial differentiation, Lagrange's equations are invariant under this transformation;[citation needed] d d t ∂ L ′ ∂
Lagrangian_mechanics
Idealized model of light
and exit pupils. The marginal and chief rays together define the Lagrange invariant, which characterizes the throughput or etendue of the optical system
Ray_(optics)
Invariant of polynomial roots
invariant instead of to resolvent equation. A Galois resolvent is a resolvent such that the resolvent invariant is linear in the roots. The Lagrange resolvent
Resolvent_(Galois_theory)
Design rule for optical systems
so that the space-bandwidth product remains constant. Lagrange invariant Smith-Helmholtz invariant Herschel's condition Abbe, Ernst (June 1881). "On the
Abbe_sine_condition
Lagrange reversion theorem Lagrange polynomial Lagrange's four-square theorem Lagrange's theorem Lagrange's theorem (group theory) Lagrange invariant
List of scientific laws named after people
List_of_scientific_laws_named_after_people
Coordinate transformation that preserves the form of Hamilton's equations
Lorentz transformation of fields. Using Euler Lagrange relation for the provided Lagrangian, the invariants of motion can be derived as: δ L − ϵ d d t F
Canonical_transformation
Topological field
\wedge \mathbf {A} } This action is diffeomorphically invariant and gauge invariant. Its Euler–Lagrange equations are F = 0 {\displaystyle \mathbf {F} =0}
BF_model
Optical system example
| = n o / n i {\displaystyle |M_{T}|=n_{o}/n_{i}} . Lagrange invariant Smith-Helmholtz invariant Abbe sine condition Herschel, John Frederick William
Herschel's_condition
Integrable rigid bodies in classical mechanics
There are however three famous cases that are integrable, the Euler, the Lagrange, and the Kovalevskaya top, which are in fact the only integrable cases
Lagrange, Euler, and Kovalevskaya tops
Lagrange,_Euler,_and_Kovalevskaya_tops
Stress case in finite deformations
strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying tensor, in terms of an invariant one during pure
Piola–Kirchhoff stress tensors
Piola–Kirchhoff_stress_tensors
Q=Q(q,p),P=P(q,p)} is a canonical transformation, then the Lagrange bracket is an invariant of the transformation, in the sense that [ u , v ] q , p =
Lagrange_bracket
Function for incompressible divergence-free flows in two dimensions
streamfunction) are defined: The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, is defined for incompressible (divergence-free)
Stream_function
Differential calculus on function spaces
that this integral is invariant with respect to changes in the parametric representation of C . {\displaystyle C.} The Euler–Lagrange equations for a minimizing
Calculus_of_variations
Application of Lagrangian mechanics to field theories
express the Lagrangian as a function on a fiber bundle, wherein the Euler–Lagrange equations can be interpreted as specifying the geodesics on the fiber bundle
Lagrangian_(field_theory)
State invariant involving qubits
In quantum information science, the concurrence is a state invariant involving qubits. The concurrence is an entanglement monotone (a way of measuring
Concurrence (quantum computing)
Concurrence_(quantum_computing)
Technique for solving hyperbolic partial differential equations
{dz}{dt}}&=c(x,y,z).\end{aligned}}\right.} A parametrization invariant form of the Lagrange–Charpit equations is: d x a ( x , y , z ) = d y b ( x , y ,
Method_of_characteristics
Group that is also a differentiable manifold with group operations that are smooth
to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of
Lie_group
Statistical test
likelihood ratio test or the Lagrange multiplier to the Wald test: Non-invariance: As argued above, the Wald test is not invariant under reparametrization
Wald_test
Subgroup invariant under conjugation
a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the
Normal_subgroup
Statement relating differentiable symmetries to conserved quantities
}}[t],t]\,dt} is invariant under brief infinitesimal variations in the dependent variables. In other words, they satisfy the Euler–Lagrange equations d d
Noether's_theorem
Branch of mathematics
these equations as well as the invariants that may be derived from them. These equations often arise as the Euler–Lagrange equations describing the equations
Differential_geometry
Partial differential equation with nonlinear terms
equations are invariant under an infinite-dimensional gauge group, and many systems of equations (such as the Einstein field equations) are invariant under diffeomorphisms
Nonlinear partial differential equation
Nonlinear_partial_differential_equation
Gauge symmetry cannot be spontaneously broken
operators that can have non-vanishing expectation values are ones that are invariant under local gauge transformations. An important implication is that gauge
Elitzur's_theorem
Transformations induced by a mathematical group
under G {\displaystyle G} is also invariant under G {\displaystyle G} , but not conversely. Every orbit is an invariant subset of X {\displaystyle X} on
Group_action
Group that is a topological space with continuous group operations
d {\displaystyle d} on G {\displaystyle G} is called left-invariant (resp. right-invariant) if and only if d ( a x 1 , a x 2 ) = d ( x 1 , x 2 ) {\displaystyle
Topological_group
Field theory coupling of charge but not higher moments
x_{i}} and t {\displaystyle t} . This Lagrangian, combined with Euler–Lagrange equation, produces the Lorentz force law m x ¨ = q E + q x ˙ × B , {\displaystyle
Minimal_coupling
Partial differential equations whose solutions are instantons
vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the Yang–Mills action functional. They have also found significant
Yang–Mills_equations
Method of solution for certain mechanical problems
variables define a foliation by invariant Lagrangian tori because the flows induced by the Poisson commuting invariants remain within their joint level
Action-angle_coordinates
Topological space that locally resembles Euclidean space
orientability (a normal invariant, also detected by homology) and genus (a homological invariant). Smooth closed manifolds have no local invariants (other than dimension)
Manifold
Low-energy trajectories in the Solar System
little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through space can be redirected
Interplanetary Transport Network
Interplanetary_Transport_Network
Formulation of the principle of stationary action
Euler–Lagrange equations for the variational problem. Trivial examples help to appreciate the use of the action principle via the Euler–Lagrange equations
Hamilton's_principle
Application of differential geometry
diffeomorphisms according to This distance provides a right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all φ ∈ Diff
Riemannian metric and Lie bracket in computational anatomy
Riemannian_metric_and_Lie_bracket_in_computational_anatomy
Principle in Bayesian statistics
moment constraints the Lagrange multipliers are determined from the solution of a convex optimization program. The invariant measure function q(x) can
Principle_of_maximum_entropy
Differential operator acting on vector bundles
first term W μ {\displaystyle W^{\mu }} vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where U ν μ {\displaystyle
Gauge_symmetry_(mathematics)
Property of certain dynamical systems
defining property of complete integrability) the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic
Integrable_system
Geometry of the location of polynomial roots
\left|{\frac {a_{0}}{a_{n}}}\right|^{1/n}\right\}.} This bound is invariant by scaling. Lagrange improved this latter bound into the sum of the two largest values
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Fundamental mechanical principles
q(t). Starting with Hamilton's principle, the local differential Euler–Lagrange equation can be derived for systems of fixed energy. The action S {\displaystyle
Action_principles
Branch of mathematics
équations by Lagrange, which in the method of Lagrange resolvents used a complex Fourier decomposition to study the solution of a cubic: Lagrange transformed
Fourier_analysis
History of a branch of mathematics
theory of algebraic equations, number theory and geometry. Joseph Louis Lagrange, Paolo Ruffini, Niels Henrik Abel and Évariste Galois were early researchers
History_of_group_theory
differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled
Geodesics as Hamiltonian flows
Geodesics_as_Hamiltonian_flows
Relativistic quantum mechanical wave equation
forming Lorentz invariant quantities. For example, the bilinear ψ † ψ {\displaystyle \psi ^{\dagger }\psi } is not Lorentz invariant, but ψ ¯ ψ {\displaystyle
Dirac_equation
Brazilian mathematician
no. 2, pp. 285–318 (2011). with A. Avila, and J.-C. Yoccoz: "SL(2,R)-invariant probability measures on the moduli spaces of translation surfaces are
Carlos_Matheus
Physics theorem for symmetries of action
_{|I|=0}^{r}(-1)^{|I|}d_{I}{\frac {\partial L}{\partial u_{I}^{\sigma }}}} are the Euler-Lagrange expressions of the Lagrangian, and the coefficients P σ I {\textstyle P_{\sigma
Noether's_second_theorem
Mathematical formulation of special and general relativity
considered later). If a system is described by a Lagrangian L, the Euler–Lagrange equations d d t ∂ L ∂ r ˙ = ∂ L ∂ r {\displaystyle {\frac {d}{dt}}{\frac
Relativistic Lagrangian mechanics
Relativistic_Lagrangian_mechanics
polynomial Lagrange polynomial Runge's phenomenon Spline (mathematics) Bernstein polynomial Characteristic polynomial Minimal polynomial Invariant polynomial
List_of_polynomial_topics
Mathematical space used to study hyperbolic geometry
Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group
Gyrovector_space
Field theory of scalar fields
can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation
Scalar_field_theory
Type of group in mathematics
the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than
Orthogonal_group
Fields giving rise to fermionic particles
are also Lorentz invariant, this leads naturally to the Lagrangian density for the Dirac field by the requirement that the Euler–Lagrange equation of the
Fermionic_field
Key constraint in some theories admitting Hamiltonian formulations
theory that admits a Hamiltonian formulation and is reparametrisation-invariant. The Hamiltonian constraint of general relativity is an important non-trivial
Hamiltonian_constraint
Formulation of classical mechanics using momenta
{p}},{\boldsymbol {q}})} , the ( n {\displaystyle n} -dimensional) Euler–Lagrange equation ∂ L ∂ q − d d t ∂ L ∂ q ˙ = 0 {\displaystyle {\frac {\partial
Hamiltonian_mechanics
Property of a mass in motion
over the variable, then the equations of motion (known as the Lagrange or Euler–Lagrange equations) are a set of N equations: d d t ( ∂ L ∂ q ˙ j ) − ∂
Momentum
Mathematical object that describes the electromagnetic field in spacetime
If one forms an inner product of the field strength tensor a Lorentz invariant is formed F μ ν F μ ν = 2 ( B 2 − E 2 c 2 ) {\displaystyle F_{\mu \nu
Electromagnetic_tensor
Physical quantity of dimension energy × time
the 1740s developed early versions of the action principle. Joseph Louis Lagrange clarified the mathematics when he invented the calculus of variations.
Action_(physics)
Mathematical group based upon a finite number of elements
of G. The theorem is named after Joseph-Louis Lagrange. This provides a partial converse to Lagrange's theorem giving information about how many subgroups
Finite_group
Class of partial differential equations
geodesic flow on infinite-dimensional Lie groups equipped with right-invariant metrics. These equations generalize classical mechanical systems, such
Euler–Arnold_equation
Value remaining constant in a dynamical system
Euler–Lagrange equations. Conservative system Lyapunov function Hamiltonian system Conservation law Noether's theorem Charge (physics) Invariant (physics)
Conserved_quantity
Italian mathematician (born 1949)
Sapienza University of Rome. He studies algebraic geometry, quantum groups, invariant theory, and mathematical physics. He was born in Rome in 1949, the son
Corrado_de_Concini
Amount of matter present in an object
classical mechanics, the inert mass of a particle appears in the Euler–Lagrange equation as a parameter m: d d t ( ∂ L ∂ x ˙ i ) = m x ¨ i . {\displaystyle
Mass
Group of flat spacetime symmetries
and the Lorentz group then produce the Poincaré group. Objects that are invariant under this group are then said to possess Poincaré invariance or relativistic
Poincaré_group
Matrix of second derivatives
{\displaystyle g(\mathbf {x} )=c,} the bordered Hessian is the Hessian of the Lagrange function Λ ( x , λ ) = f ( x ) + λ [ g ( x ) − c ] {\displaystyle \Lambda
Hessian_matrix
Integral transform useful in probability theory, physics, and engineering
equations, introducing in particular the gamma function. Joseph-Louis Lagrange was an admirer of Euler and, in his work on integrating probability density
Laplace_transform
Array of numbers describing a metric connection
with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal
Christoffel_symbols
Relation satisfied by conjugate variables in quantum mechanics
electromagnetic field, the canonical momentum p is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is p kin = p − q A {\displaystyle
Canonical commutation relation
Canonical_commutation_relation
Algebraic structure with addition, multiplication, and division
first step towards the notion of a field was made in 1770 by Joseph-Louis Lagrange, who observed that permuting the zeros x1, x2, x3 of a cubic polynomial
Field_(mathematics)
Monster and modular connection
that the first few terms in the Fourier expansion of the normalized J-invariant (sequence A014708 in the OEIS) could be expressed in terms of linear combinations
Monstrous_moonshine
Interdisciplinary field of biology
demonstration of the metric properties of the right invariant metric, the demonstration that the Euler–Lagrange equations have a well-posed initial value problem
Computational_anatomy
Condition under which an odd prime is a sum of two squares
Fermat's assertion and Euler's conjecture were established by Joseph-Louis Lagrange. This more complicated formulation relies on the fact that O − 5 {\displaystyle
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Mechanism that explains the generation of mass for gauge bosons
(i.e., by introduction of the Higgs field) written in a gauge-invariant way. The Lagrange density for the Yukawa interaction of a fermion field ψ and the
Higgs_mechanism
Mathematical operation on vectors in 3D space
space (this is why the space must be oriented). The cross product is invariant under a rotation of the basis but is changed into its opposite by an odd
Cross_product
Tensor describing energy momentum density in spacetime
of the Lagrangian density, and the Euler–Lagrange equation has been used. This is symmetric and gauge-invariant. See Einstein–Hilbert action for more information
Stress–energy_tensor
or "gaugelike" "symmetries" (i.e. flows the formulation of a theory is invariant under). It is generally agreed that flows indicate nothing more than a
Analysis_of_flows
Type of non-topological soliton
particularly transparent way of finding this solution is via the method of Lagrange multipliers. In particular, in three spatial dimensions we must minimize
Q-ball
Normalized central moments
typically by a power of the standard deviation, rendering the moment scale invariant. The shape of different probability distributions can be compared using
Standardized_moment
Simple Lie group; the automorphism group of the octonions
Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group
G2_(mathematics)
French polymath (1749–1827)
finies. This provided the first correspondence between Laplace and Lagrange. Lagrange was the senior by thirteen years, and had recently founded in his
Pierre-Simon_Laplace
Mathematical model for describing material deformation under stress
Invariants of C {\displaystyle \mathbf {C} } are often used in the expressions for strain energy density functions. The most commonly used invariants
Finite_strain_theory
Type of operator in Fourier analysis
operators turns out to be broad: general theory shows that a translation-invariant operator on a group which obeys some (very mild) regularity conditions
Multiplier_(Fourier_analysis)
Group in which each element has finite order
multiple of the orders of the elements. For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent that
Torsion_group
Physics algorithm
part, of the hyperbolic network of invariant manifolds associated to the Lyapunov orbits about the L1, L2 Lagrange points near P2. The explicit determination
Weak_stability_boundary
Statistical property quantifying how much a collection of data is spread out
statistical dispersion have the useful property that they are location-invariant and linear in scale. This means that if a random variable X {\displaystyle
Statistical_dispersion
Interpretation of electrodynamics
electromagnetic field. Rather, the whole theory is encapsulated by the Lorentz-invariant action S {\displaystyle S} of particle trajectories z a μ ( τ a ) , z
Wheeler–Feynman absorber theory
Wheeler–Feynman_absorber_theory
Mathematics of smooth surfaces
Gauss, established that the Gaussian curvature is an intrinsic invariant, i.e. invariant under local isometries. This point of view was extended to higher-dimensional
Differential geometry of surfaces
Differential_geometry_of_surfaces
Theorem in calculus
ISBN 978-0-321-38700-4. In his 1762 paper on sound, Lagrange treats a special case of the divergence theorem: Lagrange (1762) "Nouvelles recherches sur la nature
Divergence_theorem
52-dimensional exceptional simple Lie group
group of automorphisms which keep the quadratic polynomials x2 + y2 + ... invariant, F4 is the group of automorphisms of the following set of 3 polynomials
F4_(mathematics)
Candidate unified theory of physics
g(x)+{\big (}D_{u}g{\big )}(x)} . Then the Euler–Lagrange equations imply that the weak Euler–Lagrange equations ∇ u ℓ | M = 0 {\displaystyle \nabla _{\mathfrak
Causal_fermion_systems
Commutative group (mathematics)
complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been
Abelian_group
Japanese mathematician
variable; its derivative is the Futaki invariant discovered a few years earlier by Akito Futaki. The Futaki invariant and Mabuchi energy are fundamental in
Toshiki_Mabuchi
Koebe 1/4 theorem (complex analysis) Lagrange inversion theorem (mathematical analysis, combinatorics) Lagrange reversion theorem (mathematical analysis
List_of_theorems
Matrix-valued random variable
unique equilibrium measure ν Q {\displaystyle \nu _{Q}} through the Euler-Lagrange variational conditions for some real constant l {\displaystyle l} 2 ∫ R
Random_matrix
Formulation of quantum mechanics
condition that determines the classical equations of motion (the Euler–Lagrange equations) is that the action has an extremum. In quantum mechanics, the
Path-integral_formulation
Advanced method of process control
that emanate from the current state and find (via the solution of Euler–Lagrange equations) a cost-minimizing control strategy until time t + T {\displaystyle
Model_predictive_control
In mathematics, invariant of square matrices
corresponding statements with respect to columns. The determinant is invariant under matrix similarity. This implies that, given a linear endomorphism
Determinant
Invariant action in bosonic string theory
The Nambu–Goto action is the simplest invariant action in bosonic string theory, and is also used in other theories that investigate string-like objects
Nambu–Goto_action
Subgroup of the group of invertible n×n matrices
of left-invariant derivations. If k is algebraically closed, a derivation D: O(G) → O(G) over k of the coordinate ring of G is left-invariant if D λ x
Linear_algebraic_group
Unified field theory
motion in the fifth dimension. The hypothesis for the metric implies an invariant five-dimensional length element d s {\displaystyle ds} : d s 2 ≡ g ~ a
Kaluza–Klein_theory
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