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Lagrangian Field Theory


Lagrangian mechanics is a powerful system for analyzing the motion of a system of particles. It can be extended to cover the dynamics of a field.

The Lagrangian function for a particle system is defined as the difference between its kinetic energy and its potential energy. The integral of the Lagrangian function over time is called the action of the system. It was initially believed that a system carried out a motion that minimized its action. This was called the Principle of Least Action. Later it was realized that the motion of a system was such that an infinitesimal variation in the path had no effect on the action of the system. The action was said to be stationary with respect to small variations in its motion.

The potential energy of a system is a function of only its space-time coordinates and the kinetic energy a function of only the time-derivatives of the coordinates. Where a system's motion is subject to constraints there exist a set of generalized coordinates that are compatible with those constraints and hence which simplify the analysis of the system's motion.

Stationarity implies that the system's motion is given by a set of equations known as the Euler-Lagrange equations for the system. For example, for a system of one particle whose potential energy depends upon only one variable x, the Euler-Lagrange equation is

((d/dt)(∂L/∂v) − (∂L/∂x) = 0
v = (dx/dt)

A beautiful result of Lagrangian mechanics is what is known as Noether's Theorem. This is that if there is a continuous transformation of the coordinates of the Lagrangian function then there exist a quantity that is conserved; i.e., is constant over time. If the Lagrangian function is unaffected by a change in the time coordinate then system energy is conserved. If it is unaffected by a change in a space coordinate then the momentum associated with that coordinate is conserved.

The momentum p associated with a coordinate x is given by

p = (∂L/∂v)
v = (dx/dt)

A comparison of this definition with the Euler-Lagrange equations reveals that

(dp/dt) = (∂L/∂x)

This leads to the Hamiltonian formulation of mechanics.


Let qμ=(q0, q1, …,qn) be the vector of generalized position coordinates. Let φ be a field variable; i.e., a variable defined over all space. The field variable could be a scalar, a vector or a tensor. In order to reduce the cumbersomeness of the equations let the derivatives of φ with respect to the spatial coordinates be denoted as ∂μφ where

μφ = (∂φ/∂qμ)

The Lagrangian density L is given by

L = L(φ, ∂μφ)

The integration of the Lagrangian density L over the spatial variables gives the Lagrangian function L for the system

L = K − V = ∫Ldμq

The integration of the Lagrangian function over time gives the action J for the field; i.e.,

J = ∫Ldt

If time is included as one of the coordinates then the action can be represented as

J = ∫Ldμ+1q

Path Variation

The Lagrangian analysis of fields is constructed in analogy with the Lagrangian analysis of particle dynamics. Therefore it is necessary to consider the Lagrangian analysis of particle dynamics. It is therefore necessary to consider the Lagrangian analysis of a particle and the simplest case will suffice.

In particle mechanics a path x'(t) that coincides with another path x(t) at the end points but differs from it elsewhere is called a variation of the path x(t) The variation is

δx(t) = x'(t) − x(t)
for 0≤t≤T

For infinitesimal variations in the path the variation in the Lagrangian density is given by:

δL = (∂L/∂x)δx + (∂L/∂v)δv
v = dx/dt


δv = δ(dx/dt) = d/dt (δx)


δL = (∂L/∂x)δx + (∂L/∂v)d/dt (δx)dt
δJ = ∫(δL)dt
and hence
δJ = ∫[(∂L/∂x)δx + (∂L/∂v)d/dt (δx)]dt

The term ∫ (∂L/∂v)d/dt (δx)]dt can be integrated by parts (∫UdV=UV−∫VdU) to

(∂L/∂v)δx − ∫(d/dt(∂L/∂v))δxdt

Since δx(t) is equal to zero at the limits of the integration the first term vanishes.


δJ = ∫[δ(∂L/∂v)δx − ∫(d/dt(∂L/∂v))δx]dt
or, equivalently
δJ = ∫[δ(∂L/∂v) − ∫(d/dt(∂L/∂v))]δxdt

Stationarity of action requires that δJ is equal to zero. Thus

δJ = ∫[δ(∂L/∂v) − ∫(d/dt(∂L/∂v))]δxdt = 0

Since the variation δx(t) is arbitrary the only way for the integral to be zero is for the integrand to be zero everywhere on the interval; i.e.,

L/∂v) − ∫(d/dt(∂L/∂v) = 0

This is the Euler-Lagrange equation for the particle's dynamics. The other important equation for the particle is the equation defining the momentum p of the particle. That equation is

p = (∂L/∂v)
v = (dx/dt)

The Lagrangian density of a field is a function of φ(qμ), which id the field variable as a function of the coordinates and the derivatives ∂μφ of the field variable with respect to the coordinates. For example, the coordinates may be time and the spatial coordinates.

In symbols the relationships are

L = L(φ, ∂μφ)
where ∂μφ
represents the vector of
partial derivatives with
respect to the coordinates

The Euler-Lagrange equation for a field is

(∂L/∂φ) − ∂μ(∂L/∂vμ) = 0
vμ = ∂μφ

and where the repetition of the index μ in the second term indicates that there is a summation of that term with respect to that index.

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