Infinitesimal amplitudes

A convenience of electronic basis functions (53) is that they reduce at infinitesimal-amplitude bending to (28) with the same meaning of the angle 9 we may employ these asymptotic forms in the computation of the matrix elements of the kinetic energy operator and in this way avoid the necessity of carrying out calculations of the derivatives of the electronic wave functions with respect to the nuclear coordinates. The electronic part of the Hamiltonian is represented in the basis (53) by... 

The same situation exists in polyatomic molecules. The true force constant for classical vibrations of infinitesimal amplitude... 

In the perfectly harmonic crystal the total vibrational energy equals the sum of the energies of the simple harmonic oscillators or normal modes which have infinitesimal amplitude and ignore each other. The introduction of anharmonic coupling between oscillators leads initially to small shifts in... 

The equations (12-20)-( 12-24) are the so-called linear stability equations for this problem in the inviscid fluid limit. We wish to use these equations to investigate whether an arbitrary, infinitesimal perturbation will grow or decay in time. Although the perturbation has an arbitrary form, we expect that it must satisfy the linear stability equations. Thus, once we specify an initial form for one of the variables like the pressure p, we assume that the other variables take a form that is consistent with p by means of Eqs. (12-20)-(12-24). Now the obvious question is this How do we represent a disturbance function of arbitrary form For this, we take advantage of the fact that the governing equations and boundary conditions are now linear, so that we can represent any smooth disturbance function by means of a Fourier series representation. Instead of literally studying a disturbance function of arbitrary form, we study the dynamics of all of the possible Fourier modes. If any mode is found to grow with time, the system is unstable because, with a disturbance of infinitesimal amplitude, every possible mode will always be present. 

It must be remembered, however, that the above methods for calculating cellular flow patterns are valid only for infinitesimal amplitudes, because linear theory (which requires that disturbance flows grow exponentially without limit) cannot by itself predict the final steady flows which are established. On the other hand, flows of macroscopic amplitudes would require consideration of the nonlinear terms in the equations of motion and perhaps the variation of fluid properties with temperature. Still, researchers have persisted in... 

Prosperetti [6] applied an alternate technique, based on the use of Laplace transforms, to the initial value problem of infinitesimal-amplitude oscillations of viscous drops. His results show that the motion consists of modulated oscillations with varying frequency and damping parameter. The frequency of oscillations for small viscosity is given by ... 

Since the whole treatment has so far been restricted to the case of infinitesimal amplitudes of vibration, all displacements can be considered... 

It is necessary to note the essential physical difference between the system (9) and its asymptotic approximation at <5 0 that is the equation (14). The system (9) at finite values S describes the physical waves and is suitable to comparison with experiments. The asymptotic equation (14) simulates the mathematical waves of unbounded length and infinitesimal amplitude and is not included parameters connecting with experimental conditions. This circumstance defines the preference of (9) before (14). It is important to note that mathematical model for nonlinear waves is reduced to single equation in the limiting case (5 0 only. That model includes a system of two equations for finite S values. 

The uniform phase becomes unstable against a charge density wave of wavevector q and infinitesimal amplitude when Cs q) of (1.167) vanishes [59]. This instability for q 2kp arises at low density as a consequence of exchange and correlation. 

The physically possible patterns correspond to the fraction of all the stationary solutions which is stable with respect to arbitrary disturbances of infinitesimal amplitude. In the case of the Brazovskii s model all the solutions of the first... 

The variational approach is quite approximate in the sense that it allows for large anq>Iitude vibrational motion to be described by coordinates that refer to a linearized coordinate system. Such coordinates can be used in describing motions with infinitesimal amplitudes. A more accurate representation of the transition dipole moment should, therefore, involve wave functions defined in the space of curvilinear coordinates. Appropriate transformations between different coordinate spaces can firen be applied. 

If a polymer is subjected to a sinusoidal strain y of infinitesimal amplitude yo and fixed frequency m. 

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