Lagrangian (field theory)
Lagrangian (field theory)
Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
- ,
Definitions
In Lagrangian field theory, the Lagrangian as a function of generalized coordinates is replaced by a Lagrangian density, a function of the fields in the system and their derivatives, and possibly the space and time coordinates themselves. In field theory, the independent variable t is replaced by an event in spacetime (x, y, z, t) or still more generally by a point s on a manifold.
Often, a "Lagrangian density" is simply referred to as a "Lagrangian".
Scalar fields
For many scalar fields
Vector fields, tensor fields, spinor fields
The above can be generalized for vector fields, tensor fields, and spinor fields. In physics, fermions are described by spinor fields. Bosons are described by tensor fields, which include scalar and vector fields as special cases.
Action
The time integral of the Lagrangian is called the action denoted by S. In field theory, a distinction is occasionally made between the Lagrangian L, of which the time integral is the action
The spatial volume integral of the Lagrangian density is the Lagrangian, in 3d
Mathematical formalism
Consider a functional,
- ,
called the action.
It is assumed below, in addition, that the Lagrangian depends on only the field value and its first derivative but not the higher derivatives.
From this we get:
Hence we get the Euler–Lagrange equations (due to the boundary conditions):
Examples
Newtonian gravity
where G in m3·kg−1·s−2 is the gravitational constant. Variation of the integral with respect to Φ gives:
Integrate by parts and discard the total integral. Then divide out by δΦ to get:
and thus
which yields Gauss's law for gravity.
Einstein gravity
The Lagrange density for general relativity in the presence of matter fields is
The last tensor is the energy momentum tensor and is defined by
Electromagnetism in special relativity
The interaction terms
Varying this with respect to ϕ, we get
which yields Gauss' law.
which yields Ampère's law.
We can then write the interaction term as
The term we are looking out for turns out to be
We have made use of the Minkowski metric to raise the indices on the EMF tensor. In this notation, Maxwell's equations are
where ε is the Levi-Civita tensor. So the Lagrange density for electromagnetism in special relativity written in terms of Lorentz vectors and tensors is
Electromagnetism in general relativity
It can be shown that this energy momentum tensor is traceless, i.e. that
If we take the trace of both sides of the Einstein Field Equations, we obtain
So the tracelessness of the energy momentum tensor implies that the curvature scalar in an electromagnetic field vanishes. The Einstein equations are then
Additionally, Maxwell's equations are
One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given by Kaluza-Klein theory.
Electromagnetism using differential forms
Here, A stands for the electromagnetic potential 1-form, J is the current 1-form, F is the field strength 2-form and the star denotes the Hodge star operator. This is exactly the same Lagrangian as in the section above, except that the treatment here is coordinate-free; expanding the integrand into a basis yields the identical, lengthy expression. Note that with forms, an additional integration measure is not necessary because forms have coordinate differentials built in. Variation of the action leads to
These are Maxwell's equations for the electromagnetic potential. Substituting F = dA immediately yields the equation for the fields,
because F is an exact form.
Dirac Lagrangian
The Lagrangian density for a Dirac field is:[7]
Quantum electrodynamic Lagrangian
The Lagrangian density for QED is:
Quantum chromodynamic Lagrangian
See also
Calculus of variations
Covariant classical field theory
Einstein–Maxwell–Dirac equations
Functional derivative
Functional integral
Generalized coordinates
Hamiltonian field theory
Kinetic term
Lagrangian and Eulerian coordinates
Lagrangian mechanics
Lagrangian point
Lagrangian system
Onsager–Machlup function
Principle of least action
Scalar field theory