
555Lagrangian and Hamiltonian Formulations for Continuous Systems
© 2010 Taylor & Francis Group, LLC
Using this relationship, Equation 16.79 may be rewritten in the relativistically invariant form:
d
d
d
d
d
dx
A
x
A
x
µ
µ
ν
ν
µ
µ
−
=
∑
.
Alternatively, if we dene an antisymmetric second-order tensor F
μν
by
F
A
x
A
x
µν
µ
ν
ν
µ
d
d
d
d
,
the eld equations take the form
d
d
F
x
µν
µ
µ
=
∑
.
It can be shown that these equations may be deduced from Hamilton’s principle:
δδLx Lxxxx
dd
ddddd
4
1 234
0
with
LF
A
x
A
x
6
2
2
==−
∑∑∑
()
,
µν
νµ
µ
ν
ν
µ
µν
d
d
d
d
.
Other types of vector eld have also been considered. In particular, it has been shown that, as
in the scalar case, complex eld components ...