Multiply periodic systems

For the purposes of fixing the stationary states we have up to this point only considered simply or multiply periodic systems. However the general solution of the equations frequently yield motions of a more complicated character. In such a case the considerations previously discussed are not consistent with the existence and stability of stationary states whose energy is fixed with the same exactness as in multiply periodic systems. But now in order to give an account of the properties of the elements, we are forced to assume that the atoms, in the absence of external forces at any rate always possess sharp stationary states, although the general solution of the equations of motion for the atoms with several electrons exhibits no simple periodic properties of the type mentioned (Bohr [1923]). 

In classical theory particle coordinates in a multiply-periodic system may be written as a Fourier sum of harmonic terms, of the form... 

General Multiply Periodic Systems. Uniqueness of the Action Variables... 

For the purposes of fixing the stationary states we have up to this point only considered simply or multiply periodic systems. However the general solution of the equations frequently yield motions of a more complicated character. In... 

A system is said to be singly or multiply periodic if the variables just defined can be found such that each rectangular co-ordinate is periodic m the quantities Wk i.e. can be represented as a Fourier series... 

Before we can proceed to apply our results to systems of several degrees of freedom we must introduce the conception of multiply periodic functions, and examine some of their properties. 

We will now prove that the variables wk, 3k, introduced in this way, have similar properties to w and J for one degree of freedom, namely, that the qks are multiply periodic functions of the wh s with the fundamental period system... 

Since the functions O. in (5) depend only on the relative positions of the particles of the system with respect to one another and to the fixed axis, these relative positions will be determined also by wx. . . wf 1 while fixes the absolute position of the system. According to (6), 2irwf can be regarded as the mean value of the azimuth of the arbitrarily selected particle of the system over the motions of the relative angle variables w,. . . wf v The motions can therefore be considered as a multiply periodic relative one on which is superposed a uniform precession about the fixed axis. If H, regarded as a function of the Jk, does not depend on this precession is zero the system is then degenerate. 

A multiply periodic degenerate system may frequently be changed into a non-degenerate one by means of slight influences or variation of the conditions. We shall consider, in particular, the simple case where the Hamiltonian function involves a parameter A and the system is degenerate for A=0. We imagine the energy function H expanded in powers of A for sufficiently small values of A we can break off this series after the term linear in A and write... 

While the coexistence between two limit cycles or between a limit cycle and a stable steady state is also shared by the two-variable models of fig. 12.1b and c, new modes of complex dynamic behaviour arise because of the presence of a third variable in the multiply regulated system. The coexistence between three simultaneously stable limit cycles, i.e. trirhythmicity, is the first of these. Moreover, the interaction between two instability-generating mechanisms allows the appearance of complex periodic oscillations, of the bursting type, as well as chaos. The system also displays the property of final state sensitivity (Grebogi et ai, 1983a) when two stable limit cycles are separated by a regime of unstable chaos. 

Moreover, in the 1920s, Paneth drew on the metaphysical essence of elements as basic substances in order to save the periodic system firom a major crisis. Over a short period of time, many new isotopes of the elements had been discovered, such that the number of atoms or most fundamental units suddenly seemed to have multiplied. The question was whether the periodic system should continue to accommodate the traditionally regarded atoms of each element or whether it would be restructured to accommodate the more elementary isotopes that might now be taken to constitute the true atoms. Paneth s response was that the periodic system should continue as it had before, in that it should accommodate the traditional chemical atoms and not the individual isotopes of the elements.Paneth regarded isotopes as simple substances in that they are characterized by their atomic weights, while elements as basic substances are characterized in his scheme by atomic number alone. ... 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

The high temperature superconductivity discovered in the Bi-Sr-Ca-Cu-O system was found to be related to a homologous series of compounds with an idealized formulation Bi2Sr2Ca iCu 02n+4 (n = 1 to 3 or more). The n = I, 2, 3 phases have Tc values of 10-20, 90 and 11 K, respectively. The presence of superstructure modulations which are, in general, incommensurate with the basic structure was first discovered by HRTEM and ED (Shaw et al 1988, Gai et al 1988). The periodicity of the modulation is found to be about 4.7 multiplied by the a-lattice parameter. The compounds can be prepared from solid state reactions of the component oxides in stoichiometric proportions and heating between 800-900 °C in air. 

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