Lagrangian equation corrections

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

B B. By correctly choosing the coefficient matrices (a u, a 7, By, and B ), (6.159) can be made to correspond with the Eulerian velocity, composition PDF transport equation (6.19). However, it is important to note that /L / Thus it remains to determine how the Lagrangian notional-particle PDF /,( is related to the Eulerian velocity, composition PDF fn,0. This can be done by considering Lagrangian fluid particles. 

We shall first find that Lagrangian for a system of charged particles in an electromagnetic field which, through the principle of least action, gives the correct equation of motion. The electric and magnetic fields of the electromagnetic field, d and B, respectively, are related to the scalar and vector potentials, and A, by the equations... 

Equations (3.17)-(3.20) are the usual classical Hamilton s equations for a particle in the vector potential (a(q), 6(q)). Equation (3.21b) for the phase of the wave function involves the Lagrangian for such a particle plus a quantum correction. The equation of motion for the imaginary part of the overall phase, (3.22), is redundant in the sense that it merely ensures continued adherence to the normalization condition... 

The formalism of thermodynamically consistent dissipative particle dynamics represents a consistent discrete model for the Lagrangian fluctuating hydrodynamics. Equation (26.45)-Equation (26.46) conserve the mass, momentum, energy and volume. The irreversible (produced) entropy S is a strictly inerting function of time in the absence of fluctuations. Thermal fluctuations represented by F, S are consistently included, which lead to an increase of the entropy and to correct for the Einstein distribution function [31]. 

The prefactor of the interaction term has been suitably chosen in order to yield the correct equation of motion. We emphasize that the 4-potential AH = AH(x(t)) represents an external field only and therefore the Lagrangian does not dynamically depend on AH. Straightforward evaluation of the two terms of the Euler-Lagrange equations as given by Eq. (3.147) yields... 

Under the condition the Lagrangian (3. 474) be invariant for the transformation (3.468), i.e., equal with (3.473), the correction Lagrangian (3.475) should be canceled, that is the higher order differential equation is provided (Putz, 2008d)... 

The potential U, in Eq. (2.52), is called the generalized potential because it depends not only on the position of the electron and on time but also on the velocity of the electron. When we insert this Lagrangian in Lagrange s equations, Eq. (2.49), we obtain Newton s second law, Eq. (2.51) [see Exercise 2.6], which proves that our choice of the generalized potential in Eq. (2.52) was indeed correct. 

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