Elementary examples

A geodesic problem requires derivation of the shortest path connecting two points in some system for which distance is defined, subject to constraints that can be either geometrical or physical in nature. The shortest path between two points in a plane follows from this theory. The problem is to minimize 

In this example, Euler s equation takes the form of the geodesic equation 

In Johann Bernoulli s problem, the brachistochrone, it is required to find the shape of a wire such that a bead slides from point 0, 0 to xi, yi in the shortest time T under the force of gravity. The energy equation mv2 = —mgy implies v = V—2gy, so that 

On combining constants into the single parameter a this implies 

The solution for a bead starting from rest at the coordinate origin is a cycloid, determined by the parametric equations x = a( p — sin 0) and y = a (cos (f — 1). This curve is generated by a point on the perimeter of a circle of radius a that rolls below the x-axis without slipping. The lowest point occurs for p = n, with xi = it a and i = —2a. By adding a constant / o to / , a can be adjusted so that the curve passes through given points x0, and xi, y.  

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Also see Furman (1981) and Haugen (1980) for some elementary examples. For a eomprehensive referenee for the determination of stress intensity faetors for a variety of geometries and loading eonditions, see Murakami (1987). 

These are the equations on which a number of approximations are carried out to obtain approximate model solutions. In particular, the Bom-Oppenheimer (BO) frame allows for a useful separation between nuclear and electronic motion [49-52]. See also Park s book where some interesting elementary examples are analyzed concerning electro-nuclear coupling effects[53]. 

Polynomial Equations Here is an elementary example that... 

An elementary example illustrates the role of symmetry in determining when a matrix element must be equal to zero. Consider the one-dimensional integral ... 

Now let us assume that the true object is zero for all subscript values less than 1. When we do the convolution, then, im must have a value of zero for m < 1 — L. When we perform the numerical convolution starting with m 1 — L, we compute zeros as we slide the spread function toward the region of finite ok. Finally, when the end of the spread function encounters the first nonzero ok, we obtain the first nonzero value of ik. We illustrate this particular sum of products with an elementary example in which L = 1 and m — 0 ... 

Let us illustrate the application of Boltzmann s formula for an elementary example isothermal mixing of ideal gases (Sidebar 5.10). For this purpose, consider a system of Na = Nb = 4 particles. For a particular partition n 4 — n of the four A-type particles between the VA and VB containers, the number of possible ways H of choosing n A-type particles and 4 — n B-type particles for the first container is given by the product of binomial coefficients... 

The direct fluorination of almost all categories of inorganic hydrides leads to fluorine-containing analogues of the hydride precursors. The fluorination of simple inorganic hydrides (47, 48) is an elementary example of such syntheses (see Fig. 9). 

Without definite examples to focus our thinking, it is easy to get entangled in a quasi-philosophical discussion of just what a mathematical model might be. To avoid this, I present a dogmatic statement on modeling and proceed to consider an elementary example, returning later to the philosophical caveats and more general considerations. 

An elementary example of diis process is the reaction of an organometallic reactant widi a ketone (or aldehyde) followed by dehydration of the resulting alcohol to die olefin. This is truly a sequential process in that the product alcohol is dehydrated in a second, independent reaction step. It suffers as a useful synthetic method because regioisomers are often formed hi die elimination step. 

Let us illustrate previous claim with an elementary example for p = 0. In Cartesian coordinates, let... 

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... 

By now we have set up the basic machinery which permits the principles of irreversible thermodynamics to be applied to problems of interest. We next illustrate the method of procedure by an elementary example. The same approach will be used in later sections, with appropriate variations on the basic theme. 

Fig. 6-11 illustrates the dissection of the solvent-induced H NMR shift into various terms using the rather elementary example of methane [3, 266]. 

An important class of reactions has as its most elementary example the... 

The elementary example above reveals a more general truth There are going to be (at least) two time scales in weakly nonlinear oscillators. We ve already met this phenomenon in Figure 7.6.1, where the amplitude of the spiral grew very slowly compared to the cycle time. An analytical method called two-timing builds in the fact of two time scales from the start, and produces better approximations... 

The set of energy eigenvalues of a given Hamiltonian, that is, the energies that characterize the stationary states of the corresponding system is called the spectrum of the Hamiltonian and plays a critical role in both equilibrium and dynamical properties of the system. Some elementary examples of single particle Hamiltonian spectra are ... 

It follows from Coulomb s law that any distribution of electrical charge creates a potential V(r) at each point r in the surrounding space. In the elementary example of a collection of point charges Qh which may be positive or negative,... 

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