ABSTRACT: We reformulated the Lagrangian density
for single fluid by using Caputo's fractional derivative, then from the fractional Euler-Lagrangian equation we obtained the equations that described the motion of single fluid in fractional form.
Such a form allows to maintain the original Einstein Lagrangian density
There are two distinct definitions of a T-tensor from a Lagrangian: (i) the so-called "canonical" or "Noether" tensor, say [tau], is a byproduct of the Euler-Lagrange equations and (ii) the "Hilbert tensor," say T, is the symmetric tensor obtained as the derivative of the Lagrangian density
with respect to variations of the (space-time) metric.
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.
However, it should be especially stressed here that expression (A2) for the Lagrangian density
makes sense only for spatial regions free of charged particles, which may include both EM radiation and bound EM field.
A Lagrangian density
per unit volume of the reference configuration is introduced in [1-4,7,8,13]
The Lagrangian density
of the (1 + 1) dimensional Chern-Simons-Higgs system is given by
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.
Consider the Dirac-limiting lagrangian density
we can choose using only the complex valued [psi], [phi] and [[gamma].
A true unified field theory must not start with an arbitrarily concocted Lagrangian density
(with merely the appearance of the metric determinant [square root of -g] together with a sum of variables inserted by hand), for this is merely a way to embed --and not construct from first principles--a variational density in an ad hoc given space (manifold).
In addition, we clearly see that [zeta] represents the lagrangian density
characterizing the background field, thus lending support to our initial hypothesis regarding the lagrangian [XI].