The application of a gauge covariant derivative to the Euler–Lagrange equation yields a shortcut to the equations of motion for a field subject to an external force. The gauge covariant derivative includes an external force as an intrinsic part of the derivative and hence simplifies Lagrangians containing tensor and gauge covariant fields. The gauge covariant derivative used in the covariant Euler–Lagrange equation is presented as an extension of the coordinate covariant derivative used in tensor analysis. Several examples provide useful demonstrations of the covariant derivative relevant to studies in general relativity and gauge theory.
REFERENCES
1.
2.
Ian J. R.
Aitchison
, An Informal Introduction to Gauge Field Theories
(Cambridge U. P.
, New York
, 1982
), Eqs. (3.1) and (3.2).3.
L. D.
Landau
and E. M.
Lifshitz
, The Classical Theory of Fields
(Pergamon
, New York
, 1975
), Eq. (32.2).4.
5.
6.
Reference 1, p.
94
.7.
8.
9.
Reference 1, p.
71
.10.
Reference 7, Eq. (2.6.20).
11.
Reference 7, Eq. (2.7.30).
12.
Reference 4, Eqs. (9.2), (9.4), (9.6), and (9.7).
13.
14.
Steven
Weinberg
, The Quantum Theory of Fields
(Cambridge U. P.
, New York
, 1996
), Vol. II, p. 6
.© 2009 American Association of Physics Teachers.
2009
American Association of Physics Teachers
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