# Lagrangian (field theory)

**LagrangianLagrangian densityLagrangian field theoryLagrangian densitiesLagrangian density functionfield theory LagrangianLagrangian formalismLagrangian formulationLagrangiansmatter Lagrangian**

Lagrangian field theory is a formalism in classical field theory.wikipedia

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### Lagrangian mechanics

**LagrangianLagrange's equationsLagrangians**

It is the field-theoretic analogue of Lagrangian mechanics.

Lagrangian mechanics applies to the dynamics of particles, while fields are described using a Lagrangian density.

### Classical field theory

**field equationsclassical field theoriesfield theory**

Lagrangian field theory is a formalism in classical field theory.

A field theory tends to be expressed mathematically by using Lagrangians.

### Scalar curvature

**Ricci scalarcurvaturecurvature scalar**

:R is the curvature scalar, which is the Ricci tensor contracted with the metric tensor, and the Ricci tensor is the Riemann tensor contracted with a Kronecker delta.

In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action.

### Action (physics)

**actionaction principleaction integral**

The dependent variables (q) are replaced by the value of a field at that point in spacetime so that the equations of motion are obtained by means of an action principle, written as:

### On shell and off shell

**on-shelloff-shellon shell**

Given boundary conditions, basically a specification of the value of \varphi at the boundary if M is compact or some limit on \varphi as x → ∞ (this will help in doing integration by parts), the subspace of \mathcal{C} consisting of functions, \varphi, such that all functional derivatives of S at \varphi are zero and \varphi satisfies the given boundary conditions is the subspace of on shell solutions.

Consider a Lagrangian density given by.

### Kaluza–Klein theory

**Kaluza–KleinKaluza–Klein theoriesKaluza-Klein**

One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given by Kaluza-Klein theory.

Next, one takes this scalar curvature as the Lagrangian density, and, from this, constructs the Einstein–Hilbert action for the bundle, as a whole.

### Hamiltonian field theory

**Covariant Hamiltonian field theoryHamiltonian densitycovariant (polysymplectic) Hamiltonian field theory**

It is a formalism in classical field theory alongside Lagrangian field theory, and has applications in quantum field theory also.

### Einstein–Hilbert action

**Einstein-Hilbert actionEinstein–Hilbert LagrangianEinstein-Hilbert Lagrangian**

The integral of is known as the Einstein-Hilbert action.

When a cosmological constant Λ is included in the Lagrangian, the action:

### Gauss's law for gravity

**Gauss's lawfor gravityGauss' law for gravity**

which yields Gauss's law for gravity.

The Lagrangian density for Newtonian gravity is

### Noether's theorem

**Noether currentNoether chargeNoether theorem**

The action of a physical system is the integral over time of a Lagrangian function (which may be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.

### Kinetic term

**kinetic**

In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields (and for nonlinear sigma models, they are not even bilinear), and usually contains two derivatives with respect to time (or space); in the case of fermions, the kinetic term usually has one derivative only.

### Stress–energy tensor

**energy–momentum tensorenergy-momentum tensorstress-energy tensor**

:The last tensor is the energy momentum tensor and is defined by

where is the nongravitational part of the action, is the nongravitational part of the Lagrangian density, and the Euler-Lagrange equation has been used.

### Electromagnetic tensor

**electromagnetic field tensorfield strength tensorelectromagnetic field strength tensor**

:Additionally, we can package the E and B fields into what is known as the electromagnetic tensor F_{\mu\nu}. where is the electromagnetic tensor, D is the gauge covariant derivative, and is Feynman notation for with where A_\sigma is the electromagnetic four-potential.

This means the Lagrangian density is

### Scalar field theory

**scalar fieldscalar fieldsscalar**

where \mathcal{L} is known as a Lagrangian density;

### Lagrangian system

**Lagrangianconfiguration spaceLagrangian density**

(or, simply, a Lagrangian) of order

### Fermionic field

**Dirac fieldfermion fieldfermionic**

The Lagrangian density for a Dirac field is:

Since linear combinations of these quantities 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 system recover the Dirac equation.

### Quantum electrodynamics

**QEDquantum electrodynamicelectromagnetic**

The Lagrangian density for QED is:

The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field is given in natural units by the real part of

### Quantum chromodynamics

**QCDQuantum Chromodynamics (QCD)quantum chromodynamic**

The Lagrangian density for quantum chromodynamics is:

The dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian.

### Gauge covariant derivative

**covariant derivative(gauge) covariant derivative**

where is the electromagnetic tensor, D is the gauge covariant derivative, and is Feynman notation for with where A_\sigma is the electromagnetic four-potential.

:and in the QED Lagrangian is therefore gauge invariant, and the gauge covariant derivative is thus named aptly.

### Gluon field strength tensor

**gluon field tensor**

n = 1, 2, ...6 counts the quark types, and is the gluon field strength tensor.

Characteristic of field theories, the dynamics of the field strength are summarized by a suitable Lagrangian density and substitution into the Euler–Lagrange equation (for fields) obtains the equation of motion for the field.

### Degrees of freedom (physics and chemistry)

**degrees of freedomdegree of freedomdegrees of freedom (**

Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom.

### Spacetime

**space-timespace-time continuumspace and time**

In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a manifold.

### Manifold

**manifoldsboundarymanifold with boundary**

In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a manifold.

### Equations of motion

**equation of motionUniformly accelerated motionSUVAT equations**

The dependent variables (q) are replaced by the value of a field at that point in spacetime so that the equations of motion are obtained by means of an action principle, written as: