Lagrangian
Density
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The
action for a field is defined as
L
is the Lagrangian density. [1] The Euler-Lagrange equations for the field
Lagrangian density are
where Ys
represents a component of a 4-vector that represents the field. It can be shown
that the Lagrangian density for the electromagnetic field is given by [2]
where Ja = (rc, J) is the 4-current (r = charge density, J = current density) and
is the Faraday
tensor. In this case Ys
= As=
(F/c,
A) is the 4-potential (F
= Coulomb potential, A = magnetic vector potential).
Substituting the expression
into Eq.
(3) gives
To put
this into a form containing the field variable Aa,
substitute Eq. (5) into Eq. (6) to give
Expanding
the quantities in parentheses gives
The next
step in forming the left hand side of Eq. (2) is to take the derivative of Eq.
(8) with respect to ¶aAb,
i.e.
Differentiate
the terms inside the bracket, using the product rule, gives
Using
the product rule to expand Eq. (10) to give
This
expression can also be simplified by noting that
Substituting
this relation into Eq. 11 gives
The
terms inside the square brackets can be rearranged to give
The
Faraday tensor, defined by Eq. (4), can now be substituted into Eq. (14) to give
The
coefficient glmgns
can now be brought inside the parentheses. Each term is then seen to be
identical to the other terms and has the value Fab.
So the entire expression inside the parentheses can be replaced by 4Fab.
Eq. (15) then becomes
Taking
the derivative of Eq. (16) gives the first term on the left side of Eq. (2)
The
second term in Eq. (2) is easily found by differentiating Eq. (6) with respect
to Aa
to give
Therefore
we have upon substituting Eq. (17) and (18) into Eq. (2), noting that Ys
= As,
yields
Which is
Maxwell's Equations in covariant form.
References:
[1] Classical
Electrodynamics – 3rd Ed., J.D. Jackson, John Wiley & Sons,
1999, page 599 Eq. (12.84)
[2] Ref. 1, Section 12.7, page 599, Eq. (12.85)
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