Classical mechanical systems

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

The results of the theory of quantum mechanics require that nuclear states have discrete energies. This is in contrast to classical mechanical systems, which can have any of a continuous range of energies. This difference is a critical fact in the appHcations of radioactivity measurements, where the specific energies of radiations are generally used to identify the origin of the radiation. Quantum mechanics also shows that other quantities have only specific discrete values, and the whole understanding of atomic and nuclear systems depends on these discrete quantities. 

The calculation of the torsional accelerations, i.e. the second time derivatives of the torsion angles, is the crucial point of a torsion angle dynamics algorithm. The equations of motion for a classical mechanical system with generalized coordinates are the Lagrange equations... 

The equilibrium distribution of soft generalized coordinates for both stiff and rigid classical mechanical systems of point particles may be written in the generic form... 

The proof involves the construction of a Liapunov function, a smooth, positive definite function that decreases along trajectories. As discussed in Section 7.2, a Liapunov function is a generalization of an energy function for a classical mechanical system—in the presence of friction or other dissipation, the energy decreases monotonically. There is no systematic way to concoct Liapunov functions, but often it is wise to try expressions involving sums of squares. 

Greiner W (2003) Classical Mechanics Systems of Particles and Hamiltonian Dynamics. Springer, New York... 

Despite the fact that we are not required to introduce it, the concept of resonance in classical mechanical systems has been found to be very useful in the description of the motion of systems which are for some reason or other conveniently described as containing interacting harmonic oscillators. It is found that a similar state of affairs exists in quantum mechanics. Quantum-mechanical systems which are conveniently considered to show resonance occur much more often, however, than resonating classical systems, and the resonance phenomenon has come to play an especially important part in the applications of quantum mechanics to chemistry. 

For a classical mechanical system of / degrees of freedom we may define a y phase space. This is a Euclidian space of 2/ dimensions, one for each configuration coordinate (configuration space). . . /, and one for each momentum coordinate (momentum space) Px - pf The state of each system in the ensemble would be given by a "representative point in the y phase space. The state of the ensemble as a whole would then be a "doud of points in the y phase space. We may also define a phase space as that of one molecule in the system. If we have N molecules in the system, then the state of the system is determined by one point in y space or a doud of iV points in space. For identical molecules, a cloud of JV points in fi space represents the same physical situation with interchange of the N points. And since there are iV different but equivalent arrangements, there are N points in y space that correspond to equivalent clouds in fi space. 

Given exact knowledge of the present state of a classical-mechanical system, we can predict its future state. However, the Heisenberg uncertainty principle shows that we cannot determine simultaneously the exact position and velocity of a microscopic particle, so the very knowledge required by classical mechanics for predicting the future motions of a system cannot be obtained. We must be content in quantum mechanics with something less than complete prediction of the exact future motion. 

Our approach to quantum mechanics will be to postulate the basic principles and then use these postulates to deduce experimentally testable consequences such as the energy levels of atoms. To describe the stMe of a system in quantum mechanics, we postulate the existence of a function of the particles coordinates called the wave function or state function Since the state will, in general, change with time, 9 is also a function of time. For a one-particle, one-dimensional system, we have ft = T (x, t). Hie wave function contains all possible information about a system, so instead of speaking of the state described by the wave function 9 we simply say the state 9." Newton s second law tells us how to find the future state of a classical-mechanical system from knowledge of its present state. To find the future state of a quantum-mechanical system from knowledge of its present state, we want an equation that tells us how the wave function changes with time. For a one-particle, one-dimensional system, this equation is postulated to be... 

The wave function contains all the information we can possibly know about the system it describes. What information does give us about the result of a measurement of the X coordinate of the particle We cannot expect to involve the definite specification of position that the state of a classical-mechanical system does. The correct answer to this question was provided by Max Born shortly after Schrodinger discovered the Schrodinger equation. Born postulated that... 

A classical mechanical system is characterized by a set of mechanical state variables velocity, elevation in a gravitational field, electrical charges, and so forth. If these variables are given and the external fields are known, the system is fully specified and its behavior at any instant of time, past or future, can be calculated. Thermodynamic systems are characterized by an additional state variable temperature. 

Ergodic Classical mechanical system in which a trajectory uniformly covers a specific surface in phase space. The physics literature utilizes this term to imply uniform coverage of the surface in phase space defined by fixed total energy. 

Integrable Also termed regular or quasiperiodic. Classical mechanical system characterized by the existence of a set of independent constants of the motion equal in number to the number of degrees of freedom of the system. Specific attributes of such systems are discussed in the text. 

Mixing Classical mechanical system that is ergodic and possesses additional properties associated with relaxation. 

Let be a quadratic Hamiltonian, then gradH(X) = tpX, for which the operator

Hamiltonian system of equations X = [(pX X], that is, X = sgrad T(X). Such equations are called the Euler equations. It turns out that among such systems there exist interesting multidimensional analogues of classical mechanical systems. 

Calculation of Topological Invariants of Certain Classical Mechanical Systems... 

On the one hand, chaos is good news. It explains the sense in which a classical mechanical system can forget where it came from. It implies that unless the initial conditions are fully tightly specified, the longer-time outcome is not totally determined. Initial classical conditions can never be truly fuUy specified because this calls for keeping an infinite number of digits for each number (position and velocity) and, in any case, quantum mechanics implies a necessary fuzziness in the classical initial conditions. Further, a real experiment will have even more... 

Grad H (1949) On the kinetic theory of ratified gases. Comm Pure Appl Math 2 331-407 Graaf GH, Scholtens H, Stamhuis EJ, Beenackers AACM (1990) Intra-particle diffusion limitations in low pressure methanol synthesis. Chem Eng Sci 45 773-783 Greiner W (2003) Classical mechanics systems of particles and hamUtonian dynamics. Springer, New York... 

The basic strategy for the QM/MM method lies in the hybrid potential in which a classical MM potential is combined with a QM one (Field et al. 1990). The energy of the system, , is calculated by solving the Schrbdinger equation with an effective Hamiltonian, for the mixed quantum mechanical and classical mechanical system... 

A classical mechanical system at equilibrium is at rest in a (local) minimum on the PES and will have a potential energy given by the minimum value, E ". A (real) quantum system will not, in general, be able to attain such an absolute potential energy minimum. This is due to the Heisenberg uncertainty principle, which prescribes that the uncertainty in the position of a particle. Ax, is related to the uncertainty in the momentum, Ap, of the particle ... 

The short time transition probabilities appearing in the trajectory weight are determined by the underlying dynamics. For a classical mechanical system evolving according to Hamilton s equations of motion. 

There are a number of different techniques used to solve classical mechanical systems that include Newtonian and Hamiltoitian mechanics. Hamiltoitian mechanics, though originally developed for classical systems, has a framework that is particularly useful in quantum mechaiucs. 

To better understand the quantum mechanical harmonic oscillator, the results of the quantum mechanical system can be compared to those for the classical mechanical system (described in Section 1.3). The classical turning point for the mass, , occurs when the energy of a given state is equal... 

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