Newton vector

The broad features of the lab-c.m. transformation can best be described by reference to a Newton vector diagram, like that shown in Figure 1.4, which represents a collision between A and BC with lab velocity vectors vA and vBC at right angles and a relative translational velocity v = vA — vBc. The center-of-mass vector is given by... 

Figure 1.4 Newton vector diagram for elastic scattering. It is assumed that Ma = Mb = Mc and that vA/vBC = (MBC/MA)1/2, For elastic collisions with lab velocities initially vA and vBC and a scattering angle 0, A and BC are found at the lab coordinates corresponding to the perimeters of the bases of the two cones.

If y is zero, d corresponds to the Newton vector. If y tends to infinity, d corresponds to the gradient direction with the null norm. 

In crossed molecular beam experiments, measurements of the product angle and speed are taken in the laboratory coordinate system (LAB). But information in the center-of-mass coordinate system is required to explain the dynamics of the scattering process. Thus, the results obtained in the laboratory coordinate system must be transformed to the center-of-mass coordinate system. We usually use the Newton vector diagram, i.e., the velocity vector diagram. 

Fig. 2.3 Newton vector diagram for reaction A + BC AB + C. Velocity, scattering angle, and the solid angle in the center-of-mass coordinate systems are U, 6, and Aw, respectively. The corresponding quantities in the laboratory coordinate system are V, 0, and Af2...

As a consequence of proposition 3 we obtain In the vicinity of a minimizer of E the Newton vectors and the steepest descent vectors always point to the minimizer. In the vicinity of a saddle point the Newton vectors always point to the saddle point whereas a steepest descent vector point to the saddle point only if E is convex along that vector. This observation forms the basis for a modified Newton-like method which looks for stationary points of prescribed type (see Sect.2.4.3). 

The vector p is called the quasi-Newton vector at the point x. If M =H(x), then p is the Newton vector. If M =I then p corresponds to... 

As mentioned above, update formulae have been developed to avoid a frequent evaluation of the Hessian matrix. They modify a given matrix by using quantities which have been employed before in the procedure (gradient and quasi-Newton vector). If a matrix (approximating H(x )) is available, then by an update formula a sequence of matrices M may be generated which can be used instead... 

Now the question arises Under what conditions are the BFGS- (0=1) and the DFP-update (0=0) the optimally conditioned updates among the positive definite updates of the Broyden s class when p is the (juasi-Newton vector ... 

By considering that p = -M g(x ) (quasi-Newton vector ), the above equation may be rewritten as follows... 

Notice, the right-hand term is essentially determined by the ratio of the gradient vector and the quasi-Newton vector. 

The assumptions of theorem 8 may be fulfilled. If the vector p is equal to the quasi-Newton vector, i.e. 

Thus a quasi-Newton method converges superlinearly if and only if the quasi-Newton vector converges in magnitude and direction to the Newton vector. Consequentlyy the gradient method (i.e. for all k in... 

If x <0 then the calculated quasi-Newton vector p is inconsistent... 

Descent methods are specific (quasi-)Newton methods which look for minimizers only. They differ from the general (quasi-)Newton methods in the line search step which is added to ensure that the procedure makes a sufficient progress in the direction to a minimizer, particularly in the case when the initial guess is far away from a solution. Line search means that at a point x the energy functional E is minimized along the (quasi-)Newton vector p, i.e. a positive value is determined such that... 

Such step-limiting is often helpful because the direction of correction provided by the Newton-Raphson procedure, that is, the relative magnitudes of the elements of the vector J G, is very frequently more reliable than the magnitude of the correction (Naphtali, 1964). In application, t is initially set to 1, and remains at this value as long as the Newton-Raphson correotions serve to decrease the norm (magnitude) of G, that is, for... 

At any geometry g.], the gradient vector having components d EjJd Q. provides the forces (F. = -d Ej l d 2.) along each of the coordinates Q-. These forces are used in molecular dynamics simulations which solve the Newton F = ma equations and in molecular mechanics studies which are aimed at locating those geometries where the F vector vanishes (i.e. tire stable isomers and transition states discussed above). 

We consider the computation of a trajectory —X t), where X t) is a vector of variables that evolve in time —f. The vector includes all the coordinates of the particles in the system and may include the velocities as well. Unless specifically indicated otherwise X (t) includes coordinates only. The usual way in which such vectors are propagated numerically in time is via a sequence of short time solutions to a differential equation. One of the differential equations of prime concern is the Newton s equation of motion ... 

The full Newton-Raphson method computes the full Hessian A of second derivatives and then computes a new guess at the 3X coordinate vector X, according to... 

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

Generalizing the Newton-Raphson method of optimization (Chapter 1) to a surface in many dimensions, the function to be optimized is expanded about the many-dimensional position vector of a point xq... 

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 creation terms embody the changes in momentum arising from external forces in accordance with Newton s second law (F = ma). The body forces arise from gravitational, electrostatic, and magnetic fields. The surface forces are the shear and normal forces acting on the fluid diffusion of momentum, as manifested in viscosity, is included in these terms. In practice the vector equation is usually resolved into its Cartesian components and the normal stresses are set equal to the pressures over those surfaces through which fluid is flowing. 

According to Newton s third law, this force should be exactly equal and opposite to the force exerted by Qb on Qa, and this is seen to be true from the elementary theory of vectors (fa - Tb = -Fb + fa) and so... 

Besides the interaction potential, an equation is also needed for describing the dynamics of the system, i.e. how the system evolves in time. In classical mechanics this is Newton s second law (F is the force, a is the acceleration, r is the position vector and m the particle mass). 

Conservation of Momentum. If the mass of a body or system of bodies remains constant, then Newton s second law can be interpreted as a balance between force and the time rate of change of momentum, momentum being a vector quantity defined as the product of the velocity of a body and its mass. 

Am(q) and Am(p) are masses which may have arbitrary values, and they are measured in kilograms. As follows from Newton s second law, mass is a quantitative measure of inertia, since with an increase of mass the rate of a change of the particle velocity for a fixed force becomes smaller. Also is the vector ... 

Equations 4.14 and 4.15 are used to evaluate the model response and the sensitivity coefficients that are required for setting up matrix A and vector b at each iteration of the Gauss-Newton method. 

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