Frequencies Of Small Oscillations Research Articles

Dynamics and stability of localized modes in nonlinear media with point defects

The soliton states localized at a point defects are investigated by using the nonlinear Schrödinger equation for various signs of the nonlinearity and for different types of defects. The quantum interpretation of these nonlinear localized modes is given in terms of bound states of a large number of Bose particles. The dynamic properties and stability of these states for different types of interaction between elementary excitations with one another and with the defect are investigated. The boundaries of the region of existence and stability of “impurity” solitons are determined depending on the “intensity” of the defect, and the frequency of small oscillations of a soliton near the defect is calculated.

The non-linear oscillations of a satellite with third-order resonance

Plane non-linear oscillations of an artificial satellite—a rigid body—about its centre of mass in an elliptical orbit of small eccentricity are considered. It is assumed that three times the frequency of small oscillations of the satellite in a circular orbit is close to the frequency of revolution of its centre of mass. Methods of classical perturbation theory are used to reduce the problem to that of a model system, described by a Hamiltonian which is characteristic for problems involving the motion of Hamiltonian systems with one degree of freedom in the case of third-order resonance. A detailed analysis of such systems is carried out. The theory of periodic Poincare motions and KAM-theory are used to transfer the results for the model system to the complete system and to apply them to the problem of satellite motion. The question of the existence, number and stability of periodic motions with period equal to three times the period of revolution of the centre of mass of the satellite in orbit is considered, depending on the inertial parameter of the satellite and the eccentricity of the orbit. It is shown that motions of the satellite beginning in a certain neighbourhood of its eccentricity oscillations are bounded, and an estimate is given for the size of that neighbourhood.

Soliton interaction in the easy-axis 1D segnetomagnets

We study solitons in systems with ferroelectric and ferromagnetic properties (segnetomagnets). Both subsystems have an easy-axis z anisotropy. We investigate the interaction of a zero spin precession frequency ( ω = 0) dynamical magnetic soliton with the ferroelectric domain wall. It is shown that there is a stable bound state between solitons below a threshold of its co-motion velocity. The threshold is a point of bifurcation, when the double-well effective potential of interaction turns into the single-well one. We calculate the frequency of small relative oscillations of solitons.

Transient and modulation dynamics of a multimode Fabry-Pérot laser

The two-mode rate equations for a laser in a Fabry-Pérot cavity are studied, taking into account the dynamics of spatial holeburning. We prove analytically that each mode intensity has a transient behavior characterized by two damped oscillation frequencies, while the total intensity displays a domain where only the largest oscillation frequency remains; this domain appears to display antiphase dynamics. These results are generalized to N lasing modes. We then analyze the response of the two-mode laser to a periodic pump modulation. For small modulation depth, the two oscillation frequencies are clearly displayed as resonances on the response curve. For deeper modulation depth, the beating of the modulation frequency and the largest internal oscillation frequency produces a subharmonic sequence to chaos. This chaotic domain disappears in a crisis when it collides with the branch of stable periodic oscillations associated with the smaller oscillation frequency which coexists with the subharmonic cascade.

Three-body problem in the ground-state representation

The ground-state probability density of a three-body system is used to construct a classical potentialU whose minimum coincides exactly with the ground-state energy. The spectrum of excited states may approximately be obtained by imposing quasiclassical quantization conditions over the classical motion inU. We show nontrivial one-dimensional models in which either this quantization condition is exact or considerably improves the usual semiclassical quantization. For three-dimensional problems, the small-oscillation frequencies in states with total angular momentumL=0 are computed. These frequencies could represent an improvement over the frequencies of triatomic molecules computed with the use of ordinary quasiclassics for the motion of the nuclei in the molecular term. By providing a semiclassical description of the first excited quantum states, the sketched approach rises some interesting questions such as, for example, the relevance (once again) of classical chaos to quantum mechanics.

Effect of a two-dimensional potential on the rate of thermally induced escape over the potential barrier.

The thermally induced escape rate of a particle trapped in a two-dimensional (2D) potential well has been investigated through experiment and numerical simulations. The measurements were performed on a special type of superconducting quantum interference device (SQUID) which has 2 degrees of freedom. The energies associated with the motion perpendicular to (transverse) and along (longitudinal) the escape direction are quite different: the ratio between the transverse and longitudinal small oscillation frequencies is {omega}{sub {ital t}}/{omega}{sub {ital l}}{similar to}7. The SQUID's parameters, which were used to determine the potential shape and energy scales were all independently determined. All data were obtained under conditions for which the 2D thermal activation (TA) model is expected to be valid. The results were found in good agreement with the theoretical prediction. The measured thermal activation energy is found to be the same as the barrier height calculated from the independently determined potential parameters. No evidence of apparent potential barrier enhancement recently reported in a similar system was found. In addition, the results of our numerical simulations suggest that the region in which the 2D thermal activation model is applicable may be extended to barriers as low as {Delta}{ital U}{similar to}{ital k}{sub {ital B}T}.

Three-dimensional free electron laser dispersion relation including betatron oscillations

Abstract We have developed a 3-D FEL theory based upon the Maxwell-Vlasov equations including the effects of the energy spread and emittance of the electron beam, and of betatron oscillations. The radiation field is expressed in terms of Green's function of the inhomogeneous wave equation and the distribution function of the electron beam. The distribution function is expanded in terms of a set of orthogonal functions determined by the unperturbed particle distribution. The coupled Maxwell-Vlasov equations are then reduced to a matrix equation, from which a dispersion relation for the eigenvalues is derived. In the limit of small betatron oscillation frequency, the present dispersion relation reduces to the well-known cubic equation of the one-dimensional theory in the limit of large beam size, and it gives the correct gain in the limit of small beam size. Comparisons of our numerical results with other approaches show good agreement. We present a handy empirical formula for the FEL gain of a 3-D Gaussian beam, as a function of the scaled parameters, that can be used for a quick estimate of the gain.

Force between a superconductor and a permanent magnet due to trapped flux

Type-II and multiply connected type-I superconductors may experience magnetic forces that are significantly different from the well-known repulsive Meissner effect. In particular flux trapped by a superconductor may result in a force which has a stable position. We have calculated the magnetic force acting on a superconductor, based on a trapped flux model; the results have been compared with experiments at 4.2 K on (1) a block of Nb3Sn, (2) a disk of YBa2Cu3Ox, and (3) a hollow cylinder of Pb. Most of the experimental results are in reasonably good agreement with the calculations. However, an interesting discrepancy exists in the frequency of small oscillations about the stable position.

Numerical solution of axisymmetric, unsteady free-boundary problems at finite Reynolds number. II. Deformation of a bubble in a biaxial straining flow

Numerical solutions of the full Navier–Stokes equations are used to investigate the steady and unsteady deformation of a bubble in a biaxial straining flow for Reynolds numbers in the range 0≤R≤400, and Weber numbers up to O(10). The steady-state bubble shape and the frequency of small amplitude oscillations of shape are both identical for biaxial and uniaxial straining flows in the potential flow limit. However, for a large, but finite Reynolds number, the bubble shape in the biaxial straining flow is found to be fundamentally different from the shape in uniaxial flows. This is shown to be a consequence of vorticity enhancement via vortex line stretching in the biaxial flow, which does not occur in the uniaxial flow. At the highest Reynolds number considered here, R=400, the steady-state bubble behavior for low W is qualitatively similar to the potential flow case, with a limit point for existence of the low W branch of steady solutions occurring at W∼6. However, in this case a second branch of steady solutions is found for larger W≥7, which exhibits oblate bubble shapes for large W, and has no counterpart in the potential flow limit. In unsteady flows, the behavior of bubble deformation is fundamentally different in the uniaxial and biaxial flows for both high Reynolds numbers and the potential flow limit. This suggests that breakup will occur in far different ways in the two cases.

Classical Sutherland systems in the Morse potential

It is shown that the classical problem of motion of an arbitrary number of particles along a straight line with binary itneraction (shx)−2 placed under the action of the Morse potential can be solved by using the Lax representation. Exact solution of the equations of motion is reduced to calculation of the matrix exponentials. Existence conditions are determined for the points of equilibrium. The coordinates of particles at those points are defined by zeros of the generalized Laguerre polynomials, which allows us to find explicit expressions for the equilibrium-state energy and frequencies of small oscillations around equilibrium. Comparison is made with the results obtained earlier for the discrete spectrum of the corresponding quantum systems.

Internal oscillation frequencies and anharmonic effects for the double sine-Gordon kink

A simple derivation of the small oscillation frequency around 4\ensuremath{\pi}-kink solutions of the double sine-Gordon equation is presented. Small corrections to these frequencies due to anharmonic effects are also numerically and analytically investigated. The analysis is based on energetic considerations and on the mechanical interpretation of a 4\ensuremath{\pi} kink as two point particles connected by a spring.

Asymptotic motions of mechanical systems with non-holonomic constraints

The motions of mechanical systems with non-holonomic constraints close to critical points of the potential are considered. The stability of the equilibrium positions was first treated by Whittaker /1/. A theorem is given which includes earlier results /2/ as a special case, and which enables asymptotic motions to be found for new classes of potentials. Sufficient conditions are found for the equilibrium to be unstable when not all the frequencies of small oscillations vanish. Similar studies were made in /3–6/ for systems without constraints. The hypothesis can be advanced that a critical point of the potential energy is an unstable equilibrium of a mechanical system with non-holonomic constraints (linear in the velocity) when zero is not a minimum of the function V∗. Here, the origin is the equilibrium position in question, and the asterisk denotes contraction of the potential energy V to the subspace, orthogonal to all the constraints at zero. This hypothesis is proved below for the case when the MacLaurin expansion of V∗ is V∗ = V 2∗ + V k∗ + v∗ k+1 + … , where V 2∗ + V k∗ can take negative values infinitesimally close to zero ( V j∗ is a homogeneous form of degree j). This situation when V 2∗ ⩾ 0 and V k∗ ⩾ 0 is not considered. Also, to determine the absence of a minimum, higher powers must be taken into account.

Oscillations of a suspended chain

The venerable problem is considered of finding the natural frequencies of small oscillations about the equilibrium configuration of a suspended chain, under the assumptions of a continuous, perfectly flexible, inextensible, and lossless chain, supported at two fixed endpoints in a uniform gravitational field. A variety of analytical, variational, and numerical methods gives results that are valid over the full range of chain span-to-length ratios, including the limiting cases of very slack and very taut chains. Some surprising predictions of the theory are confirmed by experimental data.

Stability of a collisionless ellipsoid with oblique rotation

The equations of the nonlinear oscillations of a collisionless ellipsoid with homogeneous density are obtained. The stability of a model of an ellipsoidal star system with oblique rotation with respect to ellipsoid-ellipsoid perturbation is investigated. The region of stability has been determined by finding the characteristic frequencies of small oscillations, the equations for these being obtained by linearizing the equations of the nonlinear oscillations.

Enhanced Diffusional Separation in Liquids by Sinusoidal Oscillations

Abstract A general analysis of the problem of enhanced diffusional separation of dilute liquid solutions contained within open ended capillary tubes and subjected to axial oscillations is presented. Results show that the mass diffusion flux of the various components is equal to the product of the molecular diffusion coefficient of the species in question, the magnitude of the species axial concentration gradient, the square of the ratio of the tidal displacement to the capillary radius, and a function of the Womersley number. The earlier results of Dryer are shown to be correct for small oscillation frequencies and tube diameters, but predict effective diffusion coefficients which are too low at higher Womersley numbers. Differential diffusion separation fluxes some six orders of magnitude larger than possible with the same geometry in the absence of axial oscillations appear to be achievable for typical aqueous solutions.

Free Vibrations of Imperfect Cantilever Bars under Self-Weight Loading

This brief study examines the relationship between the natural frequency of small oscillations and the length of vertical cantilever struts in a gravitational field. The analysis uses a Rayleigh approach and includes post-buckled equilibrium states and emphasizes the influence of initial imperfections.

Postbuckling dynamics of struts as related to their loading devices

This paper examines the linear dynamics of axially compressed pin-ended ‘perfect’ inextensional struts. Relationships are established between the natural frequencies of small oscillations about any equilibrium configuration (unbuckled and postbuckled) and axial compression for systems characterized by a stable-symmetric point of bifurcation. The general result that the initial slope of the postbuckling frequency squared curve is half the prebuckling value is shown to hold for both dead and semi-rigid loading, provided the appropriate control parameter is used. For the latter, this control is an end displacement rather than the actual load experienced by the structure: use of the load may of course be desirable, but the universal slope is then lost.

Asymptotic motions and the inversion of the lagrange-dirichlet theorem

The motions of natural mechanical systems which tend to an equilibrium position as time increases without limit are studied. The degenerate case when several frequencies of small oscillations vanish is explained. An existence theorem is proved for asymptotic trajectories on the assumption that the Maclaurin series for the potential energy has the form V 2 + V m + V m + 1 + …( V 8 is a homogeneous form of degree s) and the function V 2 + V m does not have a local minimum at the equilibrium position. We proved earlier a claim /1, 2/ about the asymptotic motions for the special case when V 2  0. This theorem is used to solve the question of the existence of asymptotic trajectories in the case of simple and unimodal singularities of the potential energy, for which “canonical” normal forms are known. Similar assertions also hold for the equilibrium positions of gradient dynamic systems. The existence of a trajectory, asymptotic to the equilibrium position, naturally implies that this position is unstable in Lyapunov's sense.

Exact Calculation of Nonlinear Orbit Properties of a Synchrotron

The trajectory of a particle in an arbitrary guide field can be computed as accurately as desired by numerical integration. This "measured" trajectory can then be fit with the phase-amplitude relation describing betatron oscillations, suitably generalized to account for nonlinear effects. Information about closed on- and off-momentum orbits, as well as the frequencies of small oscillations about them, can be obtained from such fits. Thus one can compute chromaticities exactly, among other things.

Quantum Tunneling in Dissipative Systems at Finite Temperatures

A quantum system which can tunnel out of a metastable state, and which interacts with an environment at temperature $T$, is considered. It is found that heat enhances the tunneling probability at $T=0$ by a factor $\mathrm{exp}[A(T)M{\ensuremath{\omega}}_{0}\frac{{q}_{0}^{2}}{\ensuremath{\hbar}}]$, where $M$ is the mass of the system, ${\ensuremath{\omega}}_{0}$ is the frequency of small oscillations about the metastable state, and ${q}_{0}$ is the tunneling distance. For an undamped system $A(T)$ is exponentially small, $A(T)\ensuremath{\propto}\mathrm{exp}(\frac{\ensuremath{-}\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{{k}_{B}T})$, whereas for a dissipative system $A(T)$ grows algebraically with temperature.