Frequencies Of Small Oscillations Research Articles

Quantum vs. Classical Integrability in Ruijsenaars–Schneider and Calogero–Moser Systems

The relationship (resemblance and/or contrast) between quantum and classical integrability in Ruijsenaars-Schneider (R-S) and Calogero-Moser systems is addressed. The classical Calogero and Sutherland systems (based on any root system) at equilibrium have many remarkable properties; for example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all “integer valued”. These are related to the energy eigenvalues of the quantum Calogero and Sutherland systems. Similar features and results hold for the R-S type of integrable systems based on the classical root systems.

Nonlinear tight-binding approximation for Bose-Einstein condensates in a lattice

The dynamics of Bose-Einstein condensates trapped in a deep optical lattice is governed by a discrete nonlinear equation (DNL). Its degree of nonlinearity and the intersite hopping rates are retrieved from a nonlinear tight-binding approximation taking into account the effective dimensionality of each condensate. We derive analytically the Bloch and the Bogoliubov excitation spectra, and the velocity of sound waves emitted by a traveling condensate. Within a Lagrangian formalism, we obtain Newtonian-like equations of motion of localized wavepackets. We calculate the ground-state atomic distribution in the presence of an harmonic confining potential, and the frequencies of small amplitude dipole and quadrupole oscillations. We finally quantize the DNL, recovering an extended Bose-Hubbard model.

On some properties of the multiple pendulum

This note is devoted to the plane motion of a multiple pendulum consisting of a massless, inextensible string with a number of concentrated masses attached to it. It is shown that, for a pendulum of given total length, the sum of the squared reciprocal frequencies of small oscillations does not depend on the distribution of the masses along the string. In the special case of a pendulum with equal and uniformly spaced masses, this statement is related to a property of the zeros of the Laguerre polynomials. Passing to the continuum limit, one obtains a corresponding property of the zeros of the Bessel function J0. For a multi-body pendulum, the theorem remains valid, provided each member of the pendulum is suspended at a certain point of its predecessor.

On Frequencies of Small Oscillations of Some Dynamical Systems Associated with Root Systems

In the paper by F Calogero and the author [Commun. Math. Phys. 59 (1978), 109– 116] the formula for frequencies of small oscillations of the Sutherland system (A l case) was found. In the present note the generalization of this formula for the case of arbitrary root system is given.

A disk rolling on a horizontal surface without slip

A circular disk rolling on a horizontal surface without slip is a common example of a nonholonomic problem of Lagrangian mechanics. In the literature, very few such problems have been attacked developing simple and interesting results. Here, it will be shown first that it is possible to find a steady motion of the system, which is rolling in a circle. Then it will be shown that it is possible to find the expression for the frequency of small oscillations about this steady motion, including an approximate expression for the speed required for stability of that steady motion, which is accurate for circles large in radius compared to the disk. The solution is shown to reduce to the well-known critical speed for a disk rolling in a straight line. It is then surprising to note that the critical speed for stability is slower for motion in a circle than for the straight line. This result will be developed entirely analytically which seems to be quite unusual for such a nonholonomic problem of this complexity.

Polynomials associated with equilibrium positions in Calogero Moser systems

In a previous paper (Corrigan-Sasaki), many remarkable properties of classical Calogero and Sutherland systems at equilibrium are reported. For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all integer valued. The equilibrium positions of Calogero and Sutherland systems for the classical root systems (A_r, B_r, C_r and D_r) correspond to the zeros of Hermite, Laguerre, Jacobi and Chebyshev polynomials. Here we define and derive the corresponding polynomials for the exceptional (E_6, E_7, E_8, F_4 and G_2) and non-crystallographic (I_2(m), H_3 and H_4) root systems. They do not have orthogonality but share many other properties with the above mentioned classical polynomials.

Quantum versus classical integrability in Calogero$ndash$Moser systems

Calogero–Moser systems are classical and quantum integrable multiparticle dynamics defined for any root system Δ. The quantum Calogero systems having 1/q2 potential and a confining q2 potential and the Sutherland systems with 1/sin2q potentials have 'integer' energy spectra characterized by the root system Δ. Various quantities of the corresponding classical systems, e.g. minimum energy, frequencies of small oscillations, the eigenvalues of the classical Lax pair matrices etc, at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems. To our surprise, most of these classical data are also 'integers', or they appear to be 'quantized'. To be more precise, these quantities are polynomials of the coupling constant(s) with integer coefficients. The close relationship between quantum and classical integrability in Calogero–Moser systems deserves fuller analytical treatment, which would lead to better understanding of these systems and of integrable systems in general.

Internal resonance in an autonomous Hamiltonian system close to a system with a cyclic coordinate

The motion of an autonomous Hamiltonian system with two degrees of freedom, close to a system with a cyclic coordinate, is considered. It is assumed that the generating system admits of a steady rotation, the corresponding equilibrium position of the reduced system being stable in the linear approximation. It is also assumed that there is an internal resonance in the system: the ratio of the natural frequency of small oscillations of the reduced system to the frequency of variation of the cyclic coordinate is close to an integer. Non-linear oscillations of the complete system in the neighbourhood of this steady rotation are investigated. Periodic motions are constructed and their bifurcation and stability are examined. Methods of KAM theory are used to study quasi-periodic motions of the system. As an example, the problem of the motion of a nearly dynamically symmetrical heavy rigid body along an absolutely smooth horizontal plane is investigated in the case of internal resonance.

Solution of Kramers' problem for a moderately to heavily damped elastic string.

We obtain the nucleation rate of critical droplets for an elastic string moving in a double-well potential and subject to noise and damping forces. We obtain this rate for a class of potentials that includes both the asymmetric straight phi(4) and the straight phi(6) potentials. The frequencies of small oscillations about the critical droplet are obtained from a Heun equation. We solve the Fokker-Planck equation for the phase-space probability density by projecting onto the eigenfunction basis. We present a comparison with simulations for the case of the asymmetric straight phi(4) potential.

Centrifugal-dissipative instability of Rossby vortices and their cyclonic-anticyclonic asymmetry

A new linear centrifugal-dissipative mechanism is proposed that explains the vortex asymmetry observed, in particular, in the structure of low-frequency anticyclonic Rossby vortices. It is shown that the relevant centrifugal-dissipative instability, which spontaneously breaks the chiral symmetry of the vortices, takes place only in the range ω<Ω, where ω is the frequency of small oscillations corresponding to the effective solid-body rotation of a vortex and Ω is the rotation rate of a noninertial frame of reference. The onset of the instability is associated with the existence of an optimum magnitude of the frictional force. In the vortex model based on a two-dimensional oscillator with the natural frequency ω in a noninertial reference frame rotating at the rate Ω, the instability shows up as an exponential increase in the total angular momentum. It is noted that the centrifugal dissipative instability may also manifest itself in the seismically active regions of the world.

Geometry of a chaotic layer

An analysis is made of the dependence of the geometric shape of the chaotic layer near the separatrix of a nonlinear resonance of a Hamiltonian system on the parameters of this system. A separatrix algorithmic mapping, which describes the motion near the separatrix in the presence of an asymmetric perturbation having an arbitrary degree of asymmetry. The separatrix algorithmic mapping is an algorithm containing conditional transfer instructions, is considered. An analytic procedure is derived to reduce the separatrix algorithmic mapping to the unified surface of the cross section of the initial Hamiltonian system (mapping synchronization procedure). It is observed that in the case of the high-frequency perturbation λ → +∞ (where λ is the ratio of the perturbation frequency to the frequency of small phase oscillations at resonance), the chaotic layer is subjected to strong bending in the sense that during motion near the separatrix theamplitude of the energy deviations relative to the unperturbed separatrix value is much larger than the layer width. However, the synchronized separatrix algorithmic mapping ensures an accurate representation of the phase portrait of the layer for both low and high values of the parameter λ provided that the amplitude of the perturbation is fairly small. This is demonstrated by comparing the phase portraits obtained using the synchronized separatrix algorithmic mapping with the results of direct numerical integrations of the initial Hamiltonian system.

Dynamics and stability of resonant rings in galaxies

We have developed a model for a spheroidal, ring-shaped galaxy. The stars move in a ring with an elliptical cross section at the 1: 1 frequency resonance. The shape of the cross section of the equilibrium ring depends on the oblateness of the galaxy itself, so that the ellipse of the ring cross section is radially extended when the oblateness of the galaxy is small. If the oblateness of galaxy exceeds some critical value, the ellipse cross section is extended along the Ox3 axis. The shape of the ring cross section is circular for a galaxy with critical eccentricity. The stability of the ring over a wide range of perturbations is studied. A fundamental bicubic dispersion equation for the frequencies of small oscillations of a perturbed ring is derived. Application of the model to the ring galaxy NGC 7020 shows that its ring cross section should be approximately circular. Analysis of the dispersion equation demonstrates that stellar orbits in the arm are unstable (but the instability increment is small). We conclude that stars in the ring of this galaxy should drift from the 1: 1 resonance, and the ring itself should evolve.

Motion of compactonlike kinks.

We analyze the ability of a compactonlike kink (i.e., kink with compact support) to execute a stable ballistic propagation in a discrete Klein-Gordon system with anharmonic coupling. We demonstrate that the effects of lattice discreteness, and the presence of a linear coupling between lattice sites, are detrimental to a stable ballistic propagation of the compacton, because of the particular structure of the small-oscillation frequency spectrum of the compacton in which the lower-frequency internal modes enter in direct resonance with phonon modes. Our study reveals the parameter regions for obtaining a stable ballistic propagation of a compactonlike kink. Finally we investigate the interactions between compactonlike kinks.

Stochastic resonance between limit cycles. Spring pendulum in a thermostat

The effect of white noise on phase synchronization is studied numerically for a classical model of a spring pendulum with a multiple ratio of the frequencies of small oscillations (Vitt-Gorelik model). It is shown that in the model investigated a Fermi resonance regime occurs for a system in a thermostat. A new type of nonlinear dynamics is found — stochastic resonance between limit cycles.

Investigation of Unsteady Boundary Layer Developed on a Rotationally Oscillating Circular Cylinder

The unsteady boundary layer developed on a rotationally oscillating circular cylinder was examined using closely spaced, multiple hot-e lm sensor (MHFS) arrays at selected forcing rotational oscillation frequencies (Sf =foD/U0 = 0:0064‐0.0217) and amplitudes (D µ = § 27:7 and § 39:25 deg) for Re < 5:6 £ 10 4 . Hot-wire cylinderwakecharacterizationsandsmoke-tunnele ow-pattern visualizationswerealsomadetosupplement theMHFS measurements. The results show that the spatiotemporal progression of the stagnation and upper and lower separation points and the state of the unsteady boundary layer developed on the oscillating cylinder can be identie ed both nonintrusively and simultaneously. The symmetry observed in the instantaneous locations of the upper and lower separation points, at small oscillation frequency, is consistent with the identical wake width found between thestationaryandoscillatingcylinders.TheMHFSmeasurementsalsoindicateadirectcouplingbetweenthedominant frequencies of e ow oscillations at the leading-edge stagnation region, laminar-separation unsteadiness, and vortexshedding.TheMHFScharacterizationoftheunsteadyboundarylayerprovidesanincreasedunderstanding of unsteadye ow phenomena,which areprerequisites for thecontrol and management of unsteadyseparation over bluff bodies. However, higher forcing frequencies need to be investigated.

Tip Vortex Cavitation on an Oscillating Hydrofoil

This paper discusses tests conducted in the hydrodynamic tunnel of the University of Grenoble on a 3D oscillating hydrofoil. Visualization of unsteady tip vortex cavitation indicates a strong influence of the water nuclei content. The investigation was focused on the influence of the oscillation frequency on tip vortex cavitation inception. For very low nuclei content, cavitation inception is strongly delayed as compared to the steady-state results at very small oscillation frequencies. This delay is significantly reduced by nuclei seeding. The results can be explained by assuming that the time required for the inception of cavitation in the tip vortex corresponds to the time necessary for a cavitation nucleus to be captured by the vortex core.

Asymptotic equations for resonant dynamics in position-dependent magnetic fields

We consider the dynamics of a neutral atom of magnetic moment μ in a spatially dependent magnetic field. Interaction with a resonant radio-frequency field brings in non-linear dynamics for the slow motion of the center of mass. We derive asymptotically exact equations for the slow dynamics and calculate the frequency of small oscillations for resonant trapped motions.

Measuring the gravitational acceleration using a superconducting magnetic levitation system

A method is proposed for measurement of the acceleration of gravity based on the fact that, for a fixed vertical position, the frequency of small mechanical vertical oscillations of a superconducting levitated body placed in a nonuniform magnetic field is independent of the levitated mass. Particular features of the method and measurements at an uncertainty level of a few parts in 109 are discussed.

Microcosmos and Macrocosmos: A Look at these Two Universes in a Unified Way

An extension of the MIT bag model, developed to describe the strong interaction inside the hadronic matter (nucleons), is proposed as a means to account for the confinement of matter in the universe. The basic hypotheses of the MIT bag model are worked out in a very simplified way and are also translated in terms of the gravitational force. We call the nucleon "microcosmos" and the bag-universe "macrocosmos." We have found a vacuum pressure of 10-15 atm at the boundary of the bag-universe as compared with a pressure of 1029 atm at the boundary of the nucleon. Both universes are also analyzed in the light of Sciama's theory of inertia, which links the inertial mass of a body to its interaction with the rest of the universe. One of the consequences of this work is that the Weinberg mass can be interpreted as a threshold mass, namely the mass where the frequency of the small oscillations of a particle coupled to the universe matches its de Broglie frequency. Finally, we estimate an averaged density of matter in the universe, corresponding to [Formula: see text] of the critical or closure density.

Resonant cooling in position-dependent magnetic fields

We consider the dynamics of a neutral atom of magnetic moment \ensuremath{\mu} in a spatially dependent magnetic field. Interaction with a resonant radio frequency introduces a peculiar nonlinear dynamics for the slow motion of the center of mass. We derive asymptotically exact equations for the slow dynamics, which are compared to the numerical solution of the full nonlinear problem. We also calculate the frequency of small oscillations for resonant trapped motions.