Properly periodic motion

In the old (Galileo) theory of oscillations the pattern of a periodic motion was assumed to be the closed trajectory around a center. As is well known a trajectory of this kind is determined by its initial conditions—a point (x0, y0) in the phase plane. If the initial conditions are changed, there will be another closed trajectory and so on. One has, thus, a continuous family of dosed trajectories, each of which can be realized by means of proper initial conditions. 

One can employ linearly polarized light to excite selectively those fluorophores that are in a particular orientation. The difference between excitation and emitted light polarization changes whenever fluorophores rotate during the period of time between excitation and emission. The magnitude of depolarization can be measured, and one can therefore deduce the fluorophore s rotational relaxation kinetics. Extrinsic fluorescence probes are especially useful here, because the proper choice of their fluorescence lifetime will greatly improve the measurement of rotational relaxation rates. One can also determine the freedom of motion of the probe relative to the rotational diffusion properties of the macromolecule to which it is attached. When held rigidly by the macromolecule, the depolarization of a probe s fluorescence is dominated by the the motion of the macromolecule. 

In addition to stress, the other important influence on solid state kinetics (again differing from fluids) stems from the periodicity found within crystals. Crystallography defines positions in a crystal, which may be occupied by atoms (molecules) or not. If they are not occupied, they are called vacancies. In this way, a new species is defined which has attributes of the other familiar chemical species of which the crystal is composed. In normal unoccupied sublattices (properly defined interstitial lattices), the fraction of vacant sites is close to one. The motion of the atomic structure elements and the vacant lattice sites of the crystal are complementary (as is the motion of electrons and electron holes in the valence band of a semiconducting crystal). 

In this device, a thin film or periodic grating on the surface of the crystal slows the wave and prevents radiation of energy into the interior of the crystal [84]. With proper choice of crystal orientation, a purely transverse particle motion at the surface can be obtained, permitting the sensor to operate successfully in liquids. 

The M a s and Ja s we call proper angle and action variables, the wp and J/s improper or degenerate variables the w s remain constant during the motion. The number s of the independent frequencies va is called the degree of periodicity of the system. 

The idea of scaling of the environment is changing once we increase accuracy. For example, we can say that the Earth gravity at accuracy better than one-ppm level is described by three forces attraction by Earth, Sun and Moon, while the acceleration of free fall, g, is a parameter of the interaction with Earth only. Alternatively, we can say that the complete gravitational force is always mg and it is varying in time because of the relative motion of Earth, Sun and Moon. That is not only a matter of definition. It depends on natural time scale of the experiment with respect to the periods of Earth motion and on whether we understand the planetary motion properly. 

What cannot be seen in the abstract are the several themes of this short (seven-page) paper. These themes set standards both for the quality of simulation and the style of attack on complex systems. The paper discusses the methodology of numerical finite-difference integration of the dynamical equations of motion in detail in an appendix, with proper attention to numerical accuracy. The limitations of the simple pair-additive interatomic potential, of the cutoff range of the interaction, and of the periodic boundary conditions are also all noted. Validation of the underlying potential model is seriously considered via direct comparison with available experimental data for the atomic diffusion constant and for the interatomic-pair distribution function, from X-ray scattering. 

ABSTRACT. Analytical evaluation of the performance of multibody mechanical systems becomes rapidly unmanageable as the complexity of the systems increase. For problems that involve intermittent motion due to an impact, prediction of the responses is even more difficult. In an impact, nonlinear contact forces of unknown nature are created, which act and disappear over a short period of time. In this paper, different contact force models are formulated, with which a continuous analysis method is developed for a simple two-particle impact. The procedure is then generalized to impact in multibody systems using the concept of effective mass. A piecewise analysis method is discussed, which is based on a canonical form of the system impulse/momentum equations. The suitability of these methods are discussed by application of these procedures to some examples. An optimization methodology is then discussed for the selection of proper parameters in a given contact force model. The use of this technique in the selection of the most suitable materials, which are impact-resistant, is also discussed. 

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