The fractional counterpart for this Lagrangian density
[zeta] can be regarded as a Lagrangian density
characterizing a specific vacuum background field which pre-exists in the absence of matter.
In the present work, we propose another hierarchy of generalized KdV equations based on modifying the Lagrangian density
whose induced action functional is extremized by the KdV equation.
The Lagrangian is L = T - V = [[integral].sup.L.sub.0] [LAMBDA] dx in which the Lagrangian density
where L is electromagnetic Lagrangian density
. The conventional classical electrodynamics provides the Lagrange function for EM field L in explicit form as
A Lagrangian density
per unit volume of the reference configuration is introduced in [1-4,7,8,13]
The above equations show that matter Lagrangian density
and a generic function have a significant importance to discuss the dynamics of curvature-matter coupled theories.
The same applies then to the "Lagrangian density
" L = L[square root of (-g)]; that is, the latter is a smooth real function L = L([q.sup.A], [q.sup.A.sub.[mu]][[g.bar].sup.[mu]v], [[g.bar].sup.[mu]v.sub.[rho]]).
Using the phase velocity (6) and adding the interaction terms with the gauge field, we can identify the full Lagrangian density
The lagrangian density
must distinguish electrons and positron by their charge only.
However, we note that the anti-co-BRST charge [Q.sup.([bar.[lambda]]).sub.ad] (derived from the Lagrangian density
[L.sup.([bar.[lambda]]).sub.B]) and co-BRST charge [Q.sup.([lambda]).sub.d] (derived from the Lagrangian density
[L.sup.([lambda]).sub.[bar.B]]) would be different from (8).
The Lagrangian density
of the (1 + 1) dimensional Chern-Simons-Higgs system is given by