Motion equations classical

Kinetic energy The energy of motion. The classical equation for kinetic energy of a body is mv /2, where m is the mass of the body and v is its velocity. 

DIV.4. I. Prigogine and F. Henin, Radiation damping and the equation of motion in classical electrodynamics, Physica 27, 982-984 (1961). 

We introduced the equations of motion above from the Newtonian point of view using Newton s famous equation F ma. It is useful to realize that this is not the only (or even the best) way to define equations of motion within classical dynamics. Another powerful approach to this task is to define a quantity called the Lagrangian, L, in terms of the kinetic and potential energies,... 

The CMDS method is based on the Langevin equations of motion of classical PO4 dipoles ... 

The equation is defined in an I-ffame and no rotation and translation base states are present yet these latter appear after translating the origin to an I-ffame in uniform motion with (classical) velocity v, a v-frame. This aspect is not examined in this paper but observe that global angular momentum of the system is described with the invariances to rotation of the v-frame and will apply to all internal electronuclear state of the I-frame [10]. 

Cross section and potential. Collision cross sections are related to the intermolecular potential by well-known classical and quantum expressions (Hirschfelder et al, 1965 Maitland et al, 1981). Based on Newton s equation of motion the classical theory derives the expression for the scattering angle,... 

In these last two examples of equations of motion, the objective is to determine functions of the form h = /(/) or x=g(t), respectively, which satisfy the appropriate differential equation. For example, the solution of the classical harmonic motion equation is an oscillatory function, x=g t), where g(f) = cos a>t, and a> defines the frequency of oscillation. This function is represented schematically in Figure 7.1 (see also Worked Problem 4.4). 

Molecular simulations are used most often for modeling proteins and nucleic acids. We mention the methods here only because they are methods for computing a free energy directly. However, they are rather complex calculations, so we will keep our comments brief. Molecular dynamics simulations give information about the variation in structure and energy of a molecule over an interval of time (78,79). In MD, each atom moves according to Newton s equations of motion for classical particles ... 

In just the same way as in the case of fluorescence intensity, the asymptotic equations of motion of polarization moments (5.54), (5.55) and (5.87), (5.88) must coincide with the corresponding equation of motion of classical multiple moments, as introduced by Eq. (2.16). We will show that this is indeed so in the following section. 

In order to obtain the equations of motion of classical polarization moments, we must base our methods on the system of equations of motion of the probability density pa(6, angular momentum vector 3(6, optical pumping. For a number of maximally simplified situations, where the probability density in the ground state pa(6,(p) does not depend on that of the excited state pb(6,(p), we have already encountered such equations in preceding chapters see e.g., (3.4), or (4.5) and (4.6). 

Armed with these basic equations, classical partition functions and state densities for various types of motion may be evaluated either directly or via the quantum results. 

Since the classical equation of motion equates the rate of change of angular momentum J to the applied torque Mj X H, the equation of motion for the magnetization of a solid is (damping terms neglected)... 

Wall, Hiller, and Mazur [300, 301] first used a computer to integrate the classical motion equations for a system of three atoms, and in the 1960s the technique was developed by Blais and Bunker [48, 302-306], and by Karplus [19, 20, 72, 307-311] and Polanyi [71,73, 74, 267, 312, 313] and their coworkers. Recently calculations have been performed on systems simulating abstraction reactions involving more than three atoms [314] and four-center reactions involving four atoms, that is, AB + CD - AC + BD [315-317]. Here we present first a general survey of the Monte Carlo calculations of classical trajectories and then a brief review of some of the results of these calculations. Emphasis is placed on data for reactions that have been studied experimentally and have been mentioned earlier in this chapter. 

In most of the more recent classical approaches [18], no allusion to Ehrenfest s (adiabatic) principle is employed, but rather the differential equations of motion from classical mechanics are solved, either exactly or approximately, subject to a set of initial conditions (masses, force constants, interaction potential, phase, and initial energies). The amount of energy, AE, transferred to the oscillator is obtained for these conditions. This quantity may then be averaged over all phases of the oscillating molecule. In approximate classical and semiclassical treatments, the interaction potential is expanded in a Taylor s series and only the first two terms are retained. 

There are two points to note about Eqs. (61) and (67). First, as they hold for any T, Eqs. (65) and (67) provide a general restatement of the Ehrenfest relations [34], that is, the quantum analog of the equations of motion of classical mechanics. Second, they embed a series of constraints among... 

Equation (8.175) is a generalization of Ehrenfest s theorem (Ehrenfest 1927). This theorem relates the forces acting on a subsystem or atom in a molecule to the forces exerted on its surface and to the time derivative of the momentum density mJ(r). It constitutes the quantum analogue of Newton s equation of motion in classical mechanics expressed in terms of a vector current density and a stress tensor, both defined in real space. 

Electronic structure calculations of the type described above, provide the energy and related properties of the system at the absolute zero of temperature and do not account for any time-dependent effect. In some cases, temperature and/or time scale effects may be important and must be included. The appropriate theoretical approach is then molecular dynamics (MD) either in the classical or ab initio implementations. In the first approach, Newton s motion equations are solved in the field of a potential provided externally, which constitutes the main limitation of this approach. To overcome this problem, ab initio Molecular Dynamics (AIMD)94,95 solves Newton s motion equations using the ab initio potential energy surface or propagating nuclei and electrons simultaneously as in the Car-Parrinello simulation.96 The use of AIMD simulations will increase considerably in the future. In a way they furnish all the information as in classical MD, but there are no assumptions in the way the system interacts since the potential energy surface is obtained in a rather crude manner. 

A differential equation contains one or more derivatives, and its solution is a function that satisfies the equation. Classical equations of motion are differential equations based on Newton s laws of motion that when solved give the positions of particles as a function of time. We have presented the solution to several of these. These differential equations are deterministic. That is, given the equation of motion for a given system and the initial conditions (position and velocity of every particle at some initial time), the positions and velocities are determined for all times. 

The equation plays a role analogous to Newton s equation of motion in classical mechanics. In Newton s equation, the position and momentum of a particle evolve. In the time-dependent Schrddinger equation, the evolution proceeds in a completely different space-the space of states or the Hilbert space (cf.. Appendix B available at booksite.elsevier.com/978-(M44-59436-5, on p. eV). 

The MD calculation of thermodynamic functions is valid at temperatures for which the ion motion is classical. This is in the lattice-dynamics high-temperature region T > 0 cd> where is the harmonic high-temperature Debye temperature, given by equation (19.32) of Reference 33 ... 

We haven t yet considered what these vibrations might actually look like. In any system of vibrating objects, such as a molecule, there is a set of equations of motion (in classical physics) or vibrational wavefunctions (in quantum mechanics) called normal modes that describe the lowest-energy motions of the system. In the normal modes, each atom in the molecule oscillates (if it moves at all) back and forth across its equilibrium position at the same frequency and phase as every other atom in the molecule. At higher vibrational energy, the motions can be more complicated, but we can write those motions as a combination of different normal modes. Any vibration of the system can be expressed as a sum of the normal modes they are one possible basis set of vibrational coordinates. 

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