Lagrangian (field theory)

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.

Lagrangian field theory is a formalism in classical field theory.

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

: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.

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:

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.

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.

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

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

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

which yields Gauss's law for gravity.

The Lagrangian density for Newtonian gravity is

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.

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.

: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.

: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

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

(or, simply, a Lagrangian) of order

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.

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

The Lagrangian density for quantum chromodynamics is:

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

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.

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.

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

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.

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.

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: