Lagrangian derivative

It is worth investigating the time derivatives and demonstrating how to derive (9.1)-(9.4) from the more familiar forms of the conservation equations. The more familiar Lagrangian derivative djdt and d jdt are related by [9]... 

Equation (8.22) constitutes the means whereby the configuration-specific kinematic viscosity of the suspension may be computed from the prescribed spatially periodic, microscale, kinematic viscosity data v(r) by first solving an appropriate microscale unit-cell problem. Its Lagrangian derivation differs significantly from volume-average Eulerian approaches (Zuzovsky et al, 1983 Nunan and Keller, 1984) usually employed in deriving such suspension-scale properties. 

The left-hand side of the above equation is a derivative of the property in the control mass. Hence, in a control mass, the derivative is a total derivative. This is the derivative that would be observed on a given fluid property, irrespective of where the fluid is in space. Remember that the control mass system is closed no mass can enter. Therefore, the property cannot vary with space, but only with time. This derivative describes what an observer would see if travehng with the mass inside the closed container— the control mass. This is called the Lagrangian method of describing the property. This derivative is also called the Lagrangian derivative. 

In the previous equation, the total derivative is also called Lagrangian derivative, material derivative, substantive derivative, or comoving derivative. The combination of the partial derivative and the convective derivative is also called the Eulerian derivative. Again, it is very important that this equation be thoroughly understood. It is to be noted that in the enviromnental engineering literature, many authors confuse the difference between the total derivative and the partial derivative. Some authors use the partial derivative instead of the total derivative and vise versa. As shown by the previous equation, there is a big difference between the total derivative and the partial derivative. If this difference is not carefully observed, any equation written that uses one derivative instead of the other is conceptually wrong this... 

Note that [C ] is equal to zero thus, it is not appearing in Equation (9.29). It will be recalled that the left-hand side of the equation, (d[C]/dt)V, is called he Lagrangian derivative and the right-hand side exp-ressions, (d[C]/dt)V - Qo[Co] + (Qo QQ[Ce], are collectively called the Eulerian derivative ... 

At steady state, the local derivative is equal to zero. Thus, from the Reynolds transport theorem, (Lagrangian derivative = to the Eulerian derivative) ... 

Lagrangian derivative—The total rate of change of a quantity as if convection is absent in the process. 

Lagrangian derivative—The rate of change of a property when the system is closed. 

Figure 2-4. A sky diver falls with velocity Udiver from a high altitude carrying a thermometer and a recording device that plots the instantaneous temperature, as shown in the lower left-hand comer. During the period of descent, the temperature at any fixed point in the atmosphere is independent of time (i.e., the partial time derivative dT/dt =0). However, the sky diver is in an inversion layer and the temperature decreases with decreasing altitude. Thus the recording of temperature versus time obtained by the sky diver shows that the temperature decreases at a rate DT/Dt = UdiverdT/dz.. This time derivative is known as the Lagrangian derivative for an observer moving with velocity Udiver-...

The left hand side is the Lagrangian derivative of C, that is the rate of change of the concentration along a path following a fluid element. This allows to write the time evolution of the concentration C in a fluid element as ... 

The rate of change of the time-independent streamfunction along the path of a fluid element, given by its Lagrangian derivative, vanishes... 

On the left-hand side of equations (15) it is possible to notice the full Lagrangian derivative of the density of kinetic coenergy related to the fluid, adsorbate or solid component, respectively. In... 

Alternatively, a Lagrangian derivation considers a fixed mass moving with the fluid and constantly changing its volume. This method yields an alternate form of the continuity equation. 

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. 

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may oe expressed as... 

The introduction of Lagrangian coordinates in the previous section allows a more natural treatment of a continuous flow in one dimension. The derivation of the jump conditions in Section 2.2 made use of a mathematical discontinuity as a simplifying assumption. While this simplification is very useful for many applications, shock waves in reality are not idealized mathematical... 

Figure 2.12. A flow tube used to derive one-dimensional flow equations in Lagrangian coordinates. Internal surfaces are massless, impermeable partitions to aid in visualizing elements of fluid in Lagrangian coordinates.

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

The shock-change equation is the relationship between derivatives of quantities in terms of x and t (or X and t) and derivatives of variables following the shock front, which moves with speed U into undisturbed material at rest. The planar shock front is assumed to be propagating in the x (Eulerian spatial coordinate) or X (Lagrangian spatial coordinate) direction, p dx = dX. 

The conservation equations are more commonly written in the initial reference frame (Lagrangian forms). The time derivative normally used is d /dt. Equation (9.5) is used to derive (9.2) from the Lagrangian form of the conservation of mass... 

In the foregoing treatments of pressure feedback, the simulation volume retains its cubic form, so changes consist of uniform contractions and expansions. The method is readily extended to the case of a simulation region in which the lengths and directions of the edges are allowed to vary independently. Parrinello and Rahman [31] and Nose and Klein [32] extended the Andersen method to the case of noncubic simulation cells and derived a new Lagrangian for the extended system. Though their equations of motion are... 

The micromoment population balance is derived from the general population balanee (Lagrangian framework) and ean be written as... 

P. G. De Gennes. Exponents for the excluded volume problem as derived by the Wilson method. Phys Lett 38A 339, 1972 J. des Cloiseaux. The Lagrangian theory of polymer solutions at intermediate concentrations. J Phys 26 281-291, 1975. 

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... 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

The matrices B, G and are defined as partial third derivatives of the Lagrangian ... 

The response functions are obtained as derivatives of the real part of the time-averaged quasienergy Lagrangian ... 

As a consequence of the time-averaging of the quasienergy Lagrangian, the derivative in the last equation gives only a nonvanishing result if the frequencies of the external fields fulfill the matching condition Wj = 0. In fourth order Eq. (29) gives the cubic response function ... 

To derive working expressions for the dispersion coefficients Dabcd we need the power series expansion of the first-order and second-order responses of the cluster amplitudes and the Lagrangian multipliers in their frequency arguments. In Refs. [22,29] we have introduced the coupled cluster Cauchy vectors ... 

The matrices F, G, F-, H, A", B, and C which appear in the expression for the second hyperpolarizability in Eq. (30) are defined as partial derivatives of the quasienergy Lagrangian taken at zero field strengths and hence are frequency-independent. 

Interestingly, if one Taylor series expands Eq. (36) and equates the terms of the same order in kj with Eq. (37) one can derive the standard Lagrangian FD approximations (i.e., require the coefficient of kj to be —1, and require the coefficient of all other orders in kj up to the desired order of approximation to be 0.) A more global approach is to attempt to fit Eq. (36) to Eq. (37) over some range of Kj = kjA values that leads to a maximum absolute error between Eq. (36) and Eq. (37) less than or equal to some prespecrfied value, E. This is the essential idea of the dispersion-fitted finite difference method [25]. 

Ehf from equation (1-20) is obviously a functional of the spin orbitals, EHF = E[ XJ]. Thus, the variational freedom in this expression is in the choice of the orbitals. In addition, the constraint that the % remain orthonormal must be satisfied throughout the minimization, which introduces the Lagrangian multipliers e in the resulting equations. These equations (1-24) represent the Hartree-Fock equations, which determine the best spin orbitals, i. e., those (xj for which EHF attains its lowest value (for a detailed derivation see Szabo and Ostlund, 1982)... 

Using (4.26) for d/l/d(, we now have an expression for the derivative which involves the Lagrange multiplier AF and the Lagrangian ... 

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