Mechanics Newton

The fluctuations are the consequence of nondistributivity of the A transformation. We need a new mathematical framework (i.e., nondistributive algebra) to analyze nonintegrable systems. This fact reminds us that whenever we found new aspects in physics, we needed new mathematical frameworks, such as calculus for Newton mechanics, noncommutative algebra for quanmm mechanics, and the Riemann geometry for relativity. 

Second, we should consider the effect of the separation distance between molecules on the interaction forces. It was found that these fall off exponentially as the distance between atoms increases. The explanation for this behavior can be found in classical Newton mechanics according to Newton s laws, a force, F, is a push or pull exerted on a body it is a vector quantity with magnitude (newton, N = kgms 2, in the SI system) and direction. The work, W, done by the force acting on a body is given as... 

According to Newton mechanics, the infinitesimal work to pull the frame in Figure 3.5 is dW = Fx dx. Since, the acting force, F must be balanced by the force of surface tension along the length of the wire, l, of the two sides contacting the two film surfaces... 

Young-Laplace equation from Newton mechanics... 

We can derive the simplest form of the Young-Laplace equation for a spherical vapor bubble in equilibrium with liquid in a one-component system (or a liquid drop in air) from Newton mechanics. In the absence of any external field such as gravitational, magnetic or electrical fields, the bubble will assume a spherical shape, and the force acting towards the boundary of the bubble (or liquid drop) from the interior of the bubble is given as... 

Since, the work to diminish the radius of the spherical bubble is, dW7 = FydRsph from Newton mechanics, and dWr= ycLA, from Equation (189), and A = 4 Rsph, from spherical geometry, then we have... 

Since it is a very difficult task to measure d experimentally in a capillary tube, we need a relation between d and the experimentally accessible radius of the capillary tube, r. This relation can be derived by considering gravity and surface tension effects by applying fundamental Newton mechanics the complete proof is given in Section 6.1. In the case of a figure of revolution, where Rx = R2 = d, when the elevation of a general point on the surface is denoted by z, the fundamental equation is given as... 

The initial equation chosen by Lagrange to build the structure of the Newton mechanics was the equation for equilibrium of a mechanical system... 

The essential problem of the Born-Oppenheimer approximation lies in the fact, that initially the electronic states are quantized whereas the motion of nuclei remains in classical form. Then the transition from the Cartesian to the normal coordinates is carried out on the basis of Newton mechanics, and finally the nuclear motion is quantized as the system of independent harmonic oscillators. This procedure represents the hierarchical type of quantization, which is a complete contradiction of the fundamental requirement of the second quantization procedure of the total Hamiltonian that must be simultaneous. 

It is necessary to notice that the crude representation (28.30) is the first and the last one where the quantization of nuclear motion can be accomplished by means of classical Newton mechanical separations of the degrees of freedom. All other representations will mix the vibrational, rotational and translational modes, and they will not be separable any more. 

A very drastic simplification to the above-mentioned procedure to obtain the potential energy hypersurface Ui(R) is to consider the nuclei as point masses that evolve under the Newton mechanics within a conservative potential field created by the electrons. Under this classical approach the electrons do not explicitly appear and the only requirement is to have an expression for the force field. This is the theoretical basis of the MM methods. There are many different empirical force fields. They differ in the way the analytical function of the potential energy is defined and what kind of experimental data are used to fit the different parameters. They can be used for evaluating energies for systems of virtually any size so that supramolecular systems can be customarily obtained. Of course MM methods have also some severe limitations the total energy has only a relative meaning as it cannot be compared with other systems that have different number of atoms or a different structure, and the electronic effects are not considered in the MM scheme. 

Inertia forces are the uncommon forces that disobey the laws of classical Newton mechanics. Indeed, in a noninertia reference system we are unable to indicate a body whose action can explain the appearance of inertia forces. This signifies that Newtonian laws are not executed in noninertial reference systems. Figuratively speaking, there exists a force of actions (the force of inertia), but no force of counteraction. In noninertial reference systems, these particularities of inertia forces do not allow the selection of a closed system of bodies (refer to 1.3.7), since for any body in a noninertial system the inertia forces are the internal ones. Thus, in the noninertial reference system the conservation laws of energy and momentum, which will be considered below (see Section 1.5), are not valid. 

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