Lagrangian density

A few theories have an action but not a Lagrangian density.

Field equations are derived from a Lagrangian density of simple mathematical form.

The Lagrangian density is a scalar, and so will transform as .

The Lagrangian density is made of three terms:

For one scalar field , the Lagrangian density will take the form:

The Lagrangian density is slightly different from that in the previous version of the theory, and the field equations are considerably simplified.

It is a question of determining the correct Lagrangian density to generate the correct field equation.

Consider a Lagrangian density given by .

By Taylor-expanding the Lagrangian density, we can find another equivalent expression for :

Pure wave motion by linear models always leads to an averaged Lagrangian density of the form:

For an alternative formulation in terms of symmetries of the Lagrangian density, see p. 489 .

In physics, a sigma model is a physical system that is described by a Lagrangian density of the form:

The Lagrangian density for massless quarks, bound by gluons, is:

Often, a "Lagrangian density" is simply referred to as a "Lagrangian".

The Lagrangian density for a Dirac field is:

If a theory has a Lagrangian density for gravity, say , then the gravitational part of the action is the integral of that.

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

The Lagrangian is the volume integral of the Lagrangian density:

We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model Lagrangian density.

For instance for a field , described by a Lagrangian density the principle of stationary action is:

Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows:

Therefore the Lagrangian itself is equal to the integral of the Lagrangian Density over all space.

The Bretherton equation derives from the Lagrangian density:

The averaged Lagrangian density is now proposed by Whitham to follow the average variational principle:

This article uses for the Lagrangian density, and L for the Lagrangian.