Euler equations characterized

The energy and spectral optimization problems are convex programs so when there are multiple solutions the solution sets form a convex set. The following corollary characterizes how these convex sets of solutions relate to solutions of the Euler equation. In the formulation of this corollary we use the notion of optimal gap Ao—the gap achieved by optimal P and S. The optimal gap is a characteristic of the energy problem, depending only on H and S. 

Proof. By property R5 listed at the end of Section II, the elements of the Pauli subspace S are traceless, from which we infer by Theorems 12 and 13 that the energy problem and the spectral optimization problem have optimal solutions. By Theorem 10 these solutions are characterized by the Euler equation PQ = 0. ... 

The molecular dynamics constraint technique presented in the previous section is designed to simulate steady solutions of the Euler equations but there is no guarantee that all of the simulated solutions are physical. Some steady solutions are characterized by unboimded volume expansion, and others may not be the particular shock wave solutions desired. This section defines mechanical stability conditions that characterize shock waves and then shows that the molecular dynamics constraint technique naturally takes the system through states that satisfy these stability conditions. 

The Euler case. The case is characterized by the fact that a body of an arbitrary shape is supposed to be fixed in its centre of mass, that is, o o = yo = 0 = 0. In this case, the Euler equations take the form... 

Let us suppose that the liquid system is described by a MFPKE in N + 1 rigid bodies (the solute, or body 1 and N rotational solvent modes or bodies ), each characterized by inertia and friction tensors I and a set of Euler angles ft , and an angular momentum vector L (n = 1,..., N -I-1) plus K fields, each defined by a generalized mass tensor and friction tensor and a position vector and the conjugate linear momentum k = 1,..., K). The time evolution of the joint conditional probability x", L , P° 11, X, L, P, t) (where ft, X, etc. stand for the collection of Euler angles, field coordinates etc.) for the system is governed by the multivariate Fokker-Planck-Kramers equation... 

The performance of numerical methods for chemical continuity equations is generally characterized in terms of accuracy, stability, degree of mass conservation, and computational efficiency. The simplest of such methods is provided by the forward Euler or fully explicit scheme, by which the solution y" 1 at time tn y is given by... 

Now consider the change from one ground-state to another. One ground-state will be characterized by p°(r), v°(r), p°, N°, and E°, while the other one will be defined by p(r), v(r), p, N, and E. The difference between the two Euler-Lagrange equations is given by... 

The Ts[n r)] represents the kinetic energy expressed in term of the density (r). The second term of O Eq. 18.2 characterizes the interaction energy and the last one describe exchange-correlation (XC) energy Exc- In consequence the Euler-Lagrange equation may be written as so-called Kohn-Sham (SK) equation ... 

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