Ideal Resonance Research Articles

Some discussions about the ideal resonance: AAnS 32/3 (1991) 269–276

Some discussions about the ideal resonance: AAnS 32/3 (1991) 269–276

The orbital resonance among the Galilean satellites I, II and III of Jupiter

It is shown that an ideal resonance model for the first three Galilean satellites will reproduce the relevant observed facts.

Analysis of VHF Ultrasonic Resonator

The behavior of a VHF ultrasonic resonator is analyzed with a one-dimensional three layer (solid-liquid-solid) model, and resonant frequencies and the half-value width (HVW) are calculated. In contrast to an ideal liquid resonator, HVW periodically depends on the thickness of the liquid layer, which can reproduce the scattered HVW data of our previous experiment. The reason why the VHF resonator has sharper resonant peaks than the ideal liquid resonator is also explained.

A note on the application of a generalized canonical approach to non-linear Hamiltonian systems

In the author's treatment of the ideal resonance problem (1988), a non-canonical transformation was employed to bring the original Hamiltonian to a form amenable to the use of standard action-angle variables. Though the strictly Hamiltonian form of equations of motion was thus compromised, their general form was maintained, allowing transformation of the system to arbitrary order and forestalling the introduction of elliptic functions until a final explicit integration required in this approach. The general theory of such transformations is presented, and some points regarding their application are discussed, leading to the conclusion that the approach is practically limited to systems with a single degree of freedom only.

A new approach to the librational solution in the Ideal Resonance Problem

The librational motion of the Ideal Resonance Problem (Garfinkel, 1966, Jupp, 1969) is treated through an initialnon-canonical transformation which, however, leaves the equations of motion in a ‘quasi-canonical’ form, with Hamiltonian expressed in standard trigonometric functions amenable to traditional averaging techniques. The perturbed solutions, similarly expressed intrigonometric near-identity transformations, and their frequencies can be found to arbitrary order, with the elliptic integrals expected of the system introduced only in a final explicit quadrature for a Kepler-type equation in the angular variable. The specific transformations and resulting equations of motion are introduced, and explicit solutions for the original variables are found to second order, with ‘mean motion’ accurate to fifth. Limitation of the present solution to the librational region, the extension of that solution to higher order, and observations on the form of the associated Hamiltonian are also discussed.

A geometric approach to the Ideal Resonance Problem

A geometric approach to the Ideal Resonance Problem

Nonlinear dynamics in a Fabry-Perot resonator

We develop a general formalism to describe the dynamical behavior of an ensemble of two-level systems in a Fabry-Perot cavity. Our main result is a set of space and time-dependent, integro-differential equations for the slowly varying radiation and atomic variables, which we derive through a precise analysis of the slowly-varying amplitude approximation in the presence of counter-propagating fields. With the help of new and properly chosen variables we recast these equations in a form that makes their boundary conditions formally identical to those of an ideal Fabry-Perot resonator, and introduce in a natural way a modal structure even for systems with arbitrary mirror reflectivity. We derive simplified forms of these equations in the uniform field limit and within the more general single-frequency approximation. Finally, we extend our formulation to include driven systems such as optically bistable devices and the laser with an injected signal.

Computer implementation of a new approach to the ideal resonance problem

Computer implementation of a new approach to the ideal resonance problem

A ring gas laser as a quantum mechanical detector of a cooperative interaction of a radiation field with an active medium

The energy of a system composed of N two-level atoms and two counter propagating circular resonance modes in a weakly rotating toroidal ideal resonator has been found without a quantum mechanical perturbation approach. A weak rotation and a cooperative radiative interaction lead to two possible model configurations, one with nearly identical frequencies and one with different frequencies, whose difference increases with increasing rotation velocity. In the absence of rotation the perturbed frequencies (i.e. the energy of the system) are split symmetrically with respect to the unperturbed atomic frequency (i.e. the unperturbed energy of the system). A comparison of the theory with the experiment gives the estimation p_∼10 -5cm -3 for the population inversion density.

Application of the extended Delaunay method to the ideal resonance problem

In this paper, an application of the extended Delaunay methods is made to the ideal resonance problem. We show how the theory of integration proposed in a preceding paper works in a simple problem, and discuss how to proceed in more complicated situations.

The ideal resonance problem a comparison of two formal solutions III

The ideal resonance problem a comparison of two formal solutions III

NARROW WAVE BEAM-TYPE GRAVITATIONAL RADIATION OF SPACE ARRAYS

This paper touches upon gravitational radiant power and radiant angle distribution of space rectangular lattice arrays of mass quadrupole oscillators, and gives analytic expression. The calculation show that these arrays can generate narrow gravitational radiation wave beam which has the best directional property. The travelling wave typegravitational radiation theory, developed by Seki and his colleagues, may be regarded as a specific case of the work in this paper and can be derived naturally. Under the codition of ideal ultrahigh frequency acoustic resonance and perfect travelling wave type synchronism intensification, an numerical example calculated with our method shows that energy flux density of gravitational radiation of optimum direction of space array which consists of 100 CaS crystals of 30×30×0.03 cm3 would amount to 2.84×10-1 erg/cm2·s. This result shows the potentiality of crystal space arrays exciting high-frequency gravitational radiation.

Theory of feedback-generated squeezed states.

Conservation of commutator brackets imposes constraints on the response of linear systems. From these constraints the minimum uncertainties of in-phase and quadrature field components in linearized parametric and oscillator systems can be determined. In this paper, a cavity containing a ``degenerate parametric wall'' is analyzed and the resulting equations are found to be identical with those of a saturated oscillator. The ideal parametric wall resonator has lower noise than the saturated oscillator. Feedback is put onto an oscillator formed by the parametric wall, and onto the saturated oscillator, by degenerate heterodyne (homodyne) detection of the oscillator output. It is shown in both cases that the field fluctuations incident upon the photodetector in the feedback loop are ``squeezed,'' i.e., the photodetector current fluctuation level is below shot noise. A semiclassical analysis arrives at the same expression for the detector current fluctuations in the limit of a highly saturated oscillator, thus permitting an alternative interpretation of these results. However, a quantum nondemolition measurement via a nonlinear interferometer ``extracts'' the squeezed states, predicted by the quantum analysis, from the feedback loop. This result has no semiclassical interpretation.

A theory of integration for the extended Delaunay method

In this paper a method for the integration of the equations of the extended Delaunay method is proposed. It is based on the equations of the characteristic curves associated with the partial differential equation of Delaunay-Poincare. The use of the method of characteristics changes the partial differential equation for higher order approximations into a system of ordinary differential equations. The independent variable of the equations of the characteristics is used instead of the angular variables of the Jacobian methods and the averaging principle of Hori is applied to solve the equations for higher orders. It is well known that Jacobian methods applied to resonant problems generally lead to the singularity of Poincare. In the ideal resonance problem, this singularity appears when higher order approximations of the librational motion are considered. The singularity of Poincare is non-essential and is caused by the choice of the critical arguments as integration variables. The use of the independent variable of the equation of the characteristics in the place of the critical angles eliminates the singularity of Poincare.

Nonlinear and stimulated effects in resonance fluorescence of one and two atoms

The Stark representation of the resonator field and the first order of perturbation theory of the fluorescence field are used to study the fluorescence spectrum of one and two two-level atoms in an ideal resonator as a function of the initial excitation of the system. It is shown that in the absence of resonator photons (nc = 0) the spectrum obtained for initial excitation of one atom consists of two lines located symmetrically relative to the central frequency ωα. In all other cases the spectrum consists of the central harmonic ωα and of ωα ± 2Ω satellites (Ω is the Rabi frequency). The intensity of collective higher harmonics ωα ± 4Ω is low and it falls on increase in the resonator field as I/nc2. The nonlinearity of the interaction Hamiltonian results in parametric amplification of the ωα + 2Ω harmonics and in attenuation of the ωα–2Ω harmonics. In the case of two atoms it is possible to achieve stimulated fluorescence at the harmonics ωα ± 2Ω.

The ideal resonance problem, a comparison of two formal solutions, II

The ideal resonance problem, a comparison of two formal solutions, II

Studies of orbital resonance. I

view Abstract Citations (19) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Studies of orbital resonance. I Liu, L. ; Innanen, K. A. ; Zhang, S. -P. Abstract A discriminant designed to be suitable for assessing resonant stability for certain three-body configurations is described. Some aspects of the method of ideal resonance are also reviewed, emphasizing those terms in the perturbation which affect the resonance. The utility of the discriminant is illustrated by applying it to a sample of 71 asteroid orbits. Publication: The Astronomical Journal Pub Date: May 1985 DOI: 10.1086/113796 Bibcode: 1985AJ.....90..877L Keywords: Celestial Bodies; Disturbing Functions; Dynamic Stability; Orbit Perturbation; Orbital Resonances (Celestial Mechanics); Resonance; Three Body Problem; Asteroids; Discriminant Analysis (Statistics); Gravitational Constant; Hamiltonian Functions; Jupiter (Planet); Libration; Power Series; Astronomy full text sources ADS |

Resonance in regular variables I: Morphogenetic analysis of the orbits in the case of a first-order resonance

Morphogenetic analysis of the orbits of the ideal first-order resonance problem in the neighbourhood of the origin. It is shown that for problems involving central and near-central resonance it is necessary to consider as parameter the cube root of the perturbation instead of the square root used in classical non-central resonance problems.

Asymptotic solutions of the restricted problem near the equilateral Lagrangian points

The restricted problem in the vicinity of the Lagrangian point L4 is studied by finding a convergent binomial expansion of the disturbing function. Using a Hamiltonian formulation in Delaunay variables and removing the short-period terms a resonance problem (already considered by Giacaglia (1970) in an attempt of enlarging the Ideal Resonance) is obtained. It is shown that this extension is reducible to Garfinkel's ideal resonance in the libration region.

Stability in the double resonance problem

The stability of the equilibrium points and the behavior near the equilibrium points of an Ideal Double Resonance Problem are studied. In the case where the characteristic roots are purely imaginary and such that the stability cannot be decided with linear terms, the nonlinear terms are considered and some theorems of Arnold and of Khazin are used.