Lagrangian equations density

The particles position in the flow field is computed by solving the Lagrangian equations of motion for the particles with the inertial drag force, dependent on the density and size of the particles taken into account. 

We thus obtain a Lagrangean density, whieh is equivalent to Eq. (149) for all solutions of the Dirac equation, and has the structure of the nonrelativistic Lagrangian density, Eq. (140). Its variational derivations with respect to v / and v / lead to the solutions shown in Eq. (152), as well as to other solutions. 

Frieden s theory is that any physical measurement induces a transformation of Fisher information J I connecting the phenomenon being measured to intrinsic data. What we call physics - i.e. our objective description of phenomenologically observed behavior - thus derives from the Extreme Physical Information (EPI) principle, which is a variational principle. EPI asserts that, if we define K = I — J as the net physical information, K is an extremum. If one accepts this EPI principle as the foundation, the status of a Lagrangian is immediately elevated from that of a largely ad-hoc construction that yields a desired differential equation to a measure of physical information density that has a definite prior significance. 

These field equations are derivable from the following lagrangian density... 

A covariant Lagrangian density (i.e. an equation that looks the same in all Lorentz frames) will be obtained if the Lagrangian density is defined in terms of a relativistic scalar density, as in... 

The two field equations may be generated from the Lagrangian density... 

The formalism can be extended for a quantum Jield with the TFD Lagrangian density given by t = — , where is a replica of for the tilde fields so leading to similar equations of motion. For the purpose of our applications, we shall restrict our analysis to free massless fields. Thus, considering the free-massless boson (Klein-Gordon) field, the two-point Green function in the doubled space is given by... 

These equations have been obtained by Bisnovatyi-Kogan (1966), using the expression of the energy with the prescribed distributions of the density (Em-den polytrope n = 3) and entropy (arbitrary) over the Lagrangian coordinate... 

The approach described above is by no means complete or exclusive. For example, Lamb et al. (1975) have proposed an alternative route to assess the adequacy of the atmospheric diffusion equation. Their approach is based on the Lagrangian description of the statistical properties of nonreacting particles released in a turbulent atmosphere. By employing the boundary layer model of Deardorff (1970), the transition probability density p x, y, z, t x, y, z, t ) is determined from the statistics of particles released into the computed flow field. Once p has been obtained, Eq. (3.1) can then be used to derive an estimate of the mean concentration field. Finally, the validity of the atmospheric diffusion equation is assessed by determining the profile of vertical dififiisivity that produced the best fit of the predicted mean concentration field. 

Variational calculus with this Lagrangian density leads [17] to the field equation ... 

The Lagrangian (824), which is the same as the Lagrangian (839), gives the inhomogeneous equation (826) using the same Euler-Lagrange equation (843). Therefore the photon mass can be identified with the vacuum charge-current density as follows (in SI units) ... 

The Lagrangian (850) shows that 0(3) electrodynamics is consistent with the Proca equation. The inhomogeneous field equation (32) of 0(3) electrodynamics is a form of the Proca equation where the photon mass is identified with a vacuum charge-current density. To see this, rewrite the Lagrangian (850) in vector form as follows ... 

Equation (256) serves to define the Lagrangian density, L, corresponding to Euler density p. 

A major complication exists for constructing the Lagrangian density of a pair of particles diffusing relative to each other. The diffusion (Euler) equation is dissipative and the density of the diffusing species is not conserved. The Euler density, p, would lead to a space—time invariant, Sfr, which would not be constant. This difficulty requires the same approach as that used to handle the Schrodinger equation. Morse and Feshbach [499] define a reverse or backward diffusion equation where time goes backwards compared with that in eqn. (254)... 

So far, the Lagrangian density for a homogenous problem (no sink or source term in the diffusion equation) has been considered, subject to the requirement that the approximate trial function, ip, can be forced to satisfy the boundary conditions. In this sub-section, these limitations are removed and the Lagrangian density for the Green s function developed. The Green s functions for the forward and backward time process satisfy the equations... 

Equation (261) for the general Lagrangian density is rather unwieldy and unlikely to be of much direct use. Instead, the steady state version (G = G ) can be written forg = /f Gdf0... 

To develop a lower bound on the steady state, Reck and Prager [507] again considered the variational integral of eqn. (265). In this case, however, let the approximate solution j/ satisfy the diffusion equation (263) rather than the equation defining the macroscopic density M as previously done. Multiply eqn. (263) by j5(r), a Lagrangian undetermined multiplier and add it to the variational integral to give... 

The geodesic equations in the space with the metric tensor (6) can be obtained, in the usual way, by defining the Lagrangian density... 

However, these two Lagrangian densities do not couple the two fields together. This requires that the free-field equation for the gauge field becomes... 

These formulas are valid for the nonrelativistic one-electron Schrodinger equation. The Lagrangian density is... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

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