The Electromagnetic Lagrangian Density

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In applying the principle of least action we assign to a particle or a body a Lagrangian function L=T-U that we integrate with respect to time between fixed instants to calculate the particle痴 action. We then assert that the action differs not at all between the particle痴 true path and a variant path that differs from the true path by some minuscule distance:

(Eq地 1)

The calculus of variations enables us to extract from that statement the Euler-Lagrange equations, which, for a specific Lagrangian, yield up equations of motion, describing how the particle moves in space and time.

In applying the least action principle to an extended entity, a field, we assign a Lagrangian density L=L(x,A,t) such that we have

(Eq地 2)

Again we get an Euler-Lagrange equation,

(Eq地 3)

For a specific Lagrangian density that set of equations yields equations of evolution, describing how the field evolves in space and time.

Now ask whence we get the Lagrangian. If we disassemble the variational integral of Equation 1, we get the minuscule element of action, which, for a particle or body, takes the form

(Eq地 4)

How do we apply that kind of analysis to a forcefield to obtain a description of its Lagrangian density? In particular, how do we obtain the Lagrangian density of the general electromagnetic field?

Imagine the existence of a distribution of electric charge that we describe with a density function ρ(x,t). That charge distribution extends with an electromagnetic field that we describe as an electrotonic field A(x,t) and an electrostatic field φ(x,t). With respect to those potential functions the charge density has a potential linear momentum density P=ρA and a potential energy density E=ρφ. If we multiply each of those densities by a minuscule element of volume and substitute the result into the action equation, we get

(Eq地 5)

In that equation we have the contravariant four-current Ji=(ρv,ρc) and the covariant four-potential Ai=(A,φ/c).

But that equation only gives the Lagrangian density of the interaction between the field and current. Now we want to derive the Lagrangian density of the field itself.

To that end we can rewrite Equation 5 with the corresponding densities of the electromagnetic field itself. We already know how to describe the energy density of the electric and magnetic fields, so we need to devise a description of the linear momentum density that we can insert into Equation 5.

For the linear momentum density we want to calculate the amount of linear momentum that we can extract from the field through a process that extinguishes the field. The law of conservation of linear momentum guarantees that any momentum that we extract from the field must have existed in the field before we extracted it. The field in question must be entirely electric: the magnetic field doesn稚 contribute to the linear momentum, because the magnetic force only redirects linear momentum but does not produce it.

Imagine an electrically charged surface immersed in an electric field in such a way that the electric field goes to zero on one side of the surface. In that case the area density of the charge on the surface generates an electric field that augments the external field on one side of the surface and cancels it out on the other side. If the surface moves toward the electric field, we can conceive it as erasing the field as it passes over it. Using Gauss痴 law, we can calculate the area density of the charge on the surface and get σ=ε0E, in which E represents the strength of the electric field at the surface. We calculate the amount of force that the field exerts upon a minuscule patch of the surface as dF=σEda=ε0E2da, in which da represents the area of the patch. In a brief interval dt the patch moves a distance dx=vdt perpendicular to da and gains an increment of linear momentum d2P=dFdt, which it has extracted from the field in erasing a small portion of it. Thus the field must have held that much potential momentum in the element of volume dV=dadx. We can now calculate the momentum density that we assign to the field,

(Eq地 6)

That equation gives us the term that we must substitute into the version of Equation 5 through which we calculate the action density,

(Eq地 7)

So now we can combine that result with what we got from Equation 5 and write the Lagrangian density in an electromagnetic field through which electric currents flow,

(Eq地 8)

That equation shows a mixed result, part vector expression and part tensor expression. We want to make it all of one or all of the other. We can convert the interaction term back to vector form easily enough, so let痴 convert the field term into tensor form. To achieve that goal we need only note the Lorentz invariant associated with the electromagnetic field tensor,

(Eq地 9)

Now we can write the electromagnetic field痴 Lagrangian density in one of two forms, one vector and the other tensor:

(Eq地 10)

Thus we now have a technique that will enable us to devise a Lagrangian function suitable for any physical system. We now have in mind a potent tool for use in our work, because the Lagrangian encodes the laws of physics. We can even say that the Lagrangian bears to physics the same relation that DNA bears to biology: just as applying the right mix of chemicals to a strand of DNA leads to the growth of an organ suitable to the organism manifesting the DNA, so applying the Euler-Lagrange equations to any Lagrangian yields equations that describe the evolution of the physical system manifesting that Lagrangian.

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