Euler equations derivation

We continue this chapter with a presentation of the basic concepts and notations relevant to D-functional theory (Section 111). We then review the fundaments of the NOF theory (Section IV) and derive the corresponding Euler equations (Section V). The Gilbert [15] and Pernal [81] formulations, as well as the relation of Euler equations with the EKT, are considered here. The following sections are devoted to presenting our NOF theory. The cumulant of the 2-RDM is discussed in detail in Section VI. The spin-restricted formulations for closed and open-shells are analyzed in Sections Vll and VIII, respectively. Section IX is dedicated to our further simplification in order to achieve a practical functional. In Section X, we briefly describe the implementation the NOF theory for numerical calculations. We end with some results for selected molecules (Section XI). 

As in the simple 3- or 4-point bending of a beam the vibrating reed device assumes the validity of the differential Eq. (2.2) which is due to Euler. Timoshenko25 included both rotary inertia and shear deformation deriving a more exact differential equation which reduces to the Euler equation as a special case. Use of the Timoshenko beam theory for anisotropic materials has been made by Ritchie et al.26 who derive a pair of equations for torsion-flexure coupling (which will always occur unless the axis of the beam coincides with the symmetry axis of the anisotropic material). 

The first step is to derive the variational statement of the problem. This can be done with the aid of the Lagrange-Euler equation... 

Equations of state relate intensive properties to extensive properties, and are obtained from the Euler equation as partial derivatives. In the entropy representation, we have the following equations of state ... 

In contrast to fundamental equations, equations of state do not contain all the information on a system, since the intensive properties are partial derivatives of the extensive ones. To recover all the information, all the equations of state are inserted into the Euler equation. 

The Euler equation of the variation problem now involves the functional derivatives of the single-particle kinetic energy density tr corresponding to the exact many-body density p(r), and the exchange and correlation energy density exc. Such a minimization of equation (145) yields immediately... 

Having derived p exactly for 1—3 dimensions for a harmonic well, we focus on the Euler equation of the density description... 

We have demonstrated here that for the one-dimensional harmonic oscillator the integral required in the Euler equation, involving the functional derivative 8t/8p, can be exactly expressed in terms of the total kinetic energy. Indeed, the relation, involving a factor of 3, is exactly that given by the TF statistical theory. This latter theory gives for the density in d dimensions... 

There is one, admittedly elementary, example where an exact differential equation has been derived by Lawes and March.127 This is for N particles moving in a one-dimensional harmonic oscillator potential. The motivation of their argument was to study the functional derivative dtx/6p appearing in the Euler equation (49). Adapted to the linear harmonic oscillator, this reads, with suitable choice of units... 

Using the theory developed by Chapman-Enskog (see Ref. 14), a hierarchy of continuum fluid mechanics formulations may be derived from the Boltzmann equation as perturbations to the Maxwellian velocity distribution function. The first three equation sets are well known (1) the Euler equations, in which the velocity distribution is exactly the Maxwellian form (2) the Navier-Stokes equations, which represent a small deviation from Maxwellian and rely on linear expressions for viscosity and thermal conductivity and (3) the Burnett equations, which include second order derivatives for viscosity and thermal conductivity. 

Polyhedra related to the pentagonal dodecahedron and icosahedron In equation (1) for 3-connected polyhedra (p. 62) the coefficient of is zero, suggesting that polyhedra might be formed from simpler 3-connected polyhedra by adding any arbitrary number of 6-gon faces. Although such polyhedra would be consistent with equation (1) it does not follow that it is possible to construct them. The fact that a set of faces is consistent with one of the equations derived from Euler s relation does not necessarily mean that the corresponding convex polyhedron can be made. Three of the Archimedean solids are related in this way to three of the regular solids ... 

It is emphasized that the Bernoulli equation, derived from the Euler equation, is restricted to steady-, inviscid- and incompressible flow along a streamline. 

To derive the macroscopic transport equations, the conservation Relation [10] and [11] must be converted to differential equations. The main assumption needed is that the mean density and the mean velocity vary slowly in space and in time. Starting from Eq. [10] and [11], the macro dynamic equations describing the large-scale behavior of the lattice gas are obtained by multiple-scale perturbation expansion technique (Frisch et al., 1987). We shall not derive this formalism here. In the continuous limit, Eq. [10] leads to the macro dynamical conservation of mass or Euler equation... 

For comparizon, the ID version of the local Euler-Lagrange equation derived from Eq. (22),... 

The last term in Eq. (37) expresses, as before, the distortion energy, now restricted to the half-space z > 0. The Euler-Lagrange equation derived from Eq. (37) is the familiar Eq. (25) with an additional - -dependent term ip z)[asps - p z)]. 

Comparing Eqs. (33) and (34), and introducing simplifications using the Euler equation (30), one obtains the following expressions for the energy derivatives ... 

We derive the Euler equation for this problem from (5i. ..) = 0. In fact, it is the Fock equation expressed in orbitals. ... 

Newton-Euler Equations (Newton s Second Law) The equation of motion is derived using free-body diagrams (FBDs) for each rigid body. The FBDs contain kinematical (acceleration, angular acceleration, angular velocity) and dynamical (extemal/reaction forces, moments) variables. The Newton-Euler equations consist of two parts, the translational part and the rotational part. The translational part (for the ith body) is... 

Here dv/dt is the substantial derivative of the velocity or the sum of derivatives with respect to time and space. The symbols g and p denote the acceleration due to gravity and the pressure, respectively. Dealing with ideal fluids the viscosity is zero and the Navier-Stokes equation can be simplified to the Euler equation ... 

The Maxwell equations derive from the Euler reciprocity relationship ... 

The Newton-Euler (Newtonian) approach involves the derivation of the equations of motion for a dynamic system using the Newton-Euler equations, which depend upon vector quantities and accelerations. This dependence, along with complex geometries, may promote derivations for the equations of motion that are timely and mathematically complex. Furthermore, the presence of several degrees of freedom within the dynamic system will only add to the complexity of the derivations and final solutions. 

The Euler equations of motion relate the time derivative of the molecular angular velocity coj, expressed in the body-fixed (principal) axis system, to the torques given above ... 

Our task is to find an analytical expression for 9(z) at different fields. The scheme is as follows. First we shall write a proper integral equation for the free energy. Then, following the variational procedure discussed in Section 8.3, we compose the Euler equation corresponding to the free energy minimum and solve this differential equation for 9(z). To simphfy the problem we use the one-constant approximation Kii= K22 = Ks3= K.ln our geometry, and only one derivative, namely the bend term with dnjdz, is essential in the Frank free energy form (8.15) ... 

It is important to recall the generality expressed by this nonlinear relationship to help remember that the Euler equation (see Equation K6.1) is valid only in the peculiar case of linear media. From now, the relation in Equation K6.5 will be utilized in this restricted case with a scalar and constant volumic mass pfg. The surface density of percussion F,/ is linked to the previous variable according to the specificity of the particular derivative, analogous to the expression of the Lorentz force in electrodynamics (see case study F8 Hall Effect in chapter 9), and its expression can also be given in the linear case in extracting the volumic mass from the scope of the operators ... 

Formulae for the solutions are derived by successive integration of a certain system of linear inhomogeneous equations in which the Euler equations are reduced after the choice of the basis from the root vectors. A similar assertion was later announced in the paper [15]. FVom Theorem 4.2.8 there follows a useful corollary. 

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