Newton variational principle

ABSTRACT. We explore the factors responsible for the rapid convergence of the Schwinger and Newton variational principles in scattering theory. We find that, contrary to conventional wisdom, these variational methods yield high accuracy not because the error associated with the computed quantity is second oide in the error in the wavefunction, but because variational methods find wavefiinctions that are far more accurate in relevent regions of the potential, compared to nonvariational methods. 

Variational principles Newton variational principle, Schwinger variational principle, amplitude density method Truhlar, Kouri and coworkers [69], [70] ... 

The three variational principles in common use in scattering theory are due to Kohn [9], Schwinger [11] and Newton [12]. Two of these variational principles, those due to Kohn and Newton, have been successfully developed and applied to reactive scattering problems in recent years there is the S-matrix Kohn method of Zhang, Chu, and Miller, the related log derivative Kohn method of Manolopoulos, D Mello, and Wyatt and the L - Amplitude Density Generalized Newton Variational Principle (L -AD GNVP) method of Schwenke, Kouri, and Truhlar. 

The most celebrated textual embodiment of the science of energy was Thomson and Tait s Treatise on Natural Philosophy (1867). Originally intending to treat all branches of natural philosophy, Thomson and Tait in fact produced only the first volume of the Treatise. Taking statics to be derivative from dynamics, they reinterpreted Newton s third law (action-reaction) as conservation of energy, with action viewed as rate of working. Fundamental to the new energy physics was the move to make extremum (maximum or minimum) conditions, rather than point forces, the theoretical foundation of dynamics. The tendency of an entire system to move from one place to another in the most economical way would determine the forces and motions of the various parts of the system. Variational principles (especially least action) thus played a central role in the new dynamics. 

For a family of trajectories all starting at the value X(to) and at t=t all arriving at X(t), there is one trajectory that renders the action stationary. The classical mechanical trajectory of a given dynamical system is the one for which 5S=0, i.e. the action becomes stationary. The equation of motion is obtained from this variational principle [59], The corresponding Euler-Lagrange equations are obtained d(3L/3vk)/dt = 9L/dXk. In Cartesian coordinates these equations become Newton s equations of motion for each nucleus of mass Mk ... 

This is (of course) the expression for the Newton s equations of motion. The above derivation is well known and can be found in any textbook on mechanics (e.g. Landau and Lifshitz [4]). It shows the equivalence of the variation principle and the usual differential form of the equations of motion. It is also the basis for introducing initial value numerical solvers, using a finite difference to represent the second derivatives with respect to time, for example... 

R. G. Newton, Scattering Theory of Particles and Waves, 2nd ed., Springer-Verlag, New York, 1982 (Section 11.3, variational principles Section 11.2, resonances as poles of the S matrix). 

For example, even in classical mechanics we have equations (like Newton s) and the variational principle of the least action. If we introduced something similar to varying constants into the Lagrangian we will change the equations. Similar... 

Direct minimization techniques. The variational principle indicates that we want to minimize the energy as a function of the MO coefficients or the corresponding density matrix elements, as given by eq. (3.54). In this formulation, the problem is no different from other types of non-linear optimizations, and the same types of technique, such as steepest descent, conjugated gradient or Newton-Raphson methods can be used (see Chapter 12 for details). 

Minimization of the Monte Carlo energy estimate minimizes the sum of the true value and the error due to the finite sample. Although the variational principle provides a lower bound for the energy, there is no lower bound for the error of an energy estimate. Fixed sample energy minimization is therefore notoriously unstable [140, 146], Optimization algorithms based on Newton s, linear and perturbative methods have been proposed [43,48, 140,147-151]. 

As it may be hard (or even impossible) to compute the derivatives of F with respect to the manipulated variables, approximations are normally provided for both H and VF in Equations 8.25 and 8.26. (The use of numerical procedures based on variational principles was very popular in the past [ 161 ]. In order to solve variational problems numerically, standard Newton-Raphson procedures are generally used to solve the resulting two-boimdary value problem that is associated with the variational formulation. For this reason, optimization of dynamic problems based on variational principles is also included in this set of SQP-related numerical techniques.)... 

Theory. Usually we do not solve the fundamental equations directly. We use a theory, for example, Har-tree-Fock theory [3], Moller-Plesset perturbation theory [4], coupled-cluster theory [5], Kohn s [6, 7], Newton s [8], or Schlessinger s [9] variational principle for scattering amplitudes, the quasiclassical trajectory method [10], the trajectory surface hopping method [11], classical S-matrix theory [12], the close-coupling approximation... 

Variational methods are at present used extensively in the study of inelastic and reactive scattering involving atoms and diatomic molecules[l-5]. Three of the most commonly used variational methods are due to Kohn (the KVP)[6], Schwinger (the SVP)[7] and Newton (the NVP)[8]. In the applications of these methods, the wavefunction is typically expanded in a set of basis functions, parametrized by the expansion coefficients. These linear variational parameters are then determined so as to render the variational functional stationary. Unlike the variational methods in bound state calculations, the variational principles of scattering theory do not provide an upper or lower bound to the quantity of interest, except in certain special cases.[9] Neverthless, variational methods are useful because, the minimum basis size with which an acceptable level of accuracy can be achieved using a variational method is often much smaller than those required if nonvariational methods are used. The reason for this is generally explained by showing (as... 

The Kohn variational principle is perhaps the simplest of the three scattering variational principles mentioned above [9]. In particular, it requires that one calculate matrix elements only over the total Hamiltonian H of the system, and not over the Green s function Gq E) of some reference Hamiltonian Hq. While matrix elements of H between energy-independent basis functions are also energy-independent, all matrix elements of G E) have to be re-evaluated at each new scattering energy E. The Kohn variational principle is therefore somewhat easier to apply than the Schwinger and Newton... 

In the one-dimensional search methods there are two principle variations some methods employ only first derivatives of the given function (the gradient methods), whereas others (Newton s method and its variants) require explicit knowledge of the second derivatives. The methods in this last category have so far found very limited use in quantum chemistry, so that we shall refer to them only briefly at the end of this section, and concentrate on the gradient methods. The oldest of these is the method of steepest descent. 

In order to determine the distributions of pressure, velocity, and temperature the principles of conservation of mass, conservation of momentum (Newton s Law) and conservation of energy (first law of Thermodynamics) are applied. These conservation principles represent empirical models of the behavior of the physical world. They do not, of course, always apply, e.g., there can be a conversion of mass into energy in some circumstances, but they are adequate for the analysis of the vast majority of engineering problems. These conservation principles lead to the so-called Continuity, Navier-Stokes and Energy equations respectively. These equations involve, beside the basic variables mentioned above, certain fluid properties, e.g., density, p viscosity, p conductivity, k and specific heat, cp. Therefore, to obtain the solution to the equations, the relations between these properties and the pressure and temperature have to be known. (Non-Newtonian fluids in which p depends on the velocity field are not considered here.) As discussed in the previous chapter, there are, however, many practical problems in which the variation of these properties across the flow field can be ignored, i.e., in which the fluid properties can be assumed to be constant in obtaining fire solution. Such solutions are termed constant... 

In the next chapter I shall argue that it is not unlikely that Boerhaave s motto is based on Newton s first rule of reasoning in philosophy, which says that nature is pleased with simplicity, and affects not the pomp of superfluous causes. Newton, I. (1729). Mathematical Principles of Natural Philosophy and his System of the World (Andrew Motte, Trans.). London 398. Luyendijk-Elshout and Kegel-Brinkgreve did research into the source of the sentence, but did not trace an unmistakable source. However, they argue that it is a well known sentence in classic writings. They mention the example simplex est natura veritas as a variation on Boerhaave s motto. Orations. 117, n. 17. 

Following the basic principles of the Newton method, we try to find the minimum of the parametric functional in one iteration. To do so, we perturb the iteration step, Act, and find the corresponding variation of the parametric functional (11.89). According to (11.83) and (11.87), it is equal to... 

Inspired by Riemann s ideas William Clifford proposed, as a modification of Newton s universe, a space with constant positive curvature, except for small local variations. This is the most likely precursor of Einstein s closed 3-sphere universe in which matter is modelled as a pressure-less incoherent fluid, or dust, of constant density. The closure ensures constant density in space, that remains constant by defining a time coordinate orthogonal to space. The Robertson-Walker metric is defined on the same principle of constant density in a comoving coordinate system. 

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