Pendulum-like Motions Research Articles

Slow and fast collective neutrino oscillations: Invariants and reciprocity

The flavor evolution of a neutrino gas can show ''slow'' or ''fast'' collective motion. In terms of the usual Bloch vectors to describe the mean-field density matrices of a homogeneous neutrino gas, the slow two-flavor equations of motion (EOMs) are $\dot{\mathbf{P}}_\omega=(\omega\mathbf{B}+\mu\mathbf{P})\times\mathbf{P}_\omega$, where $\omega=\Delta m^2/2E$, $\mu=\sqrt{2} G_{\mathrm{F}} (n_\nu+n_{\bar\nu})$, $\mathbf{B}$ is a unit vector in the mass direction in flavor space, and $\mathbf{P}=\int d\omega\,\mathbf{P}_\omega$. For an axisymmetric angle distribution, the fast EOMs are $\dot{\mathbf{D}}_v=\mu(\mathbf{D}_0-v\mathbf{D}_1)\times\mathbf{D}_v$, where $\mathbf{D}_v$ is the Bloch vector for lepton number, $v=\cos\theta$ is the velocity along the symmetry axis, $\mathbf{D}_0=\int dv\,\mathbf{D}_v$, and $\mathbf{D}_1=\int dv\,v\mathbf{D}_v$. We discuss similarities and differences between these generic cases. Both systems can have pendulum-like instabilities (soliton solutions), both have similar Gaudin invariants, and both are integrable in the classical and quantum case. Describing fast oscillations in a frame comoving with $\mathbf{D}_1$ (which itself may execute pendulum-like motions) leads to transformed EOMs that are equivalent to an abstract slow system. These conclusions carry over to three flavors.

Symmetric and asymmetric dynamics of a tethered satellite in nontypical planes

The dynamics of tethered satellite systems operating in typical planes (e.g., the orbital plane and the planes perpendicular to the orbital plane) has been well analyzed. However, studying the orbital system in nontypical planes might be more practical. Hence, this paper concentrates on the dynamic behaviors of a tethered system with two satellites in any plane, along with active out-of-plane suppressions. First, noninertial and inertial reference frames are introduced to build a billiard and a bead model for the original system. The low-dimensional model is applicable to efficient evaluation, while the high-dimensional model is more suitable for depicting the configuration changes of a flexible tether. Then, the in-plane motion stability in a transformed reference frame is examined via the Poincaré map. A numerical dynamic zone that depends on the system parameters and initial states is utilized to sketch the system spins, pendulum-like motions, and irregular oscillations. The simulation results demonstrate that the parameter zone is periodic. The symmetry and asymmetry of the dynamic forms are also presented. Finally, a corresponding relationship between model selection and dynamic behaviors is proposed.

Phase transition and switchable dielectric behaviours in an organic–inorganic hybrid compound: (3-nitroanilinium) 2 (18-crown-6) 2 (H 2 PO 4 ) 2 (H 3 PO 4 ) 3 (H 2 O)

An organic–inorganic hybrid compound with an extensive three-dimensional (3D) crystal structure, (3-nitroanilinium)2(18-crown-6)2(H2PO4)2(H3PO4)3(H2O) (1), was synthesized under slow evaporation conditions. Differential scanning calorimetry measurements indicated that 1 underwent a reversible phase transition at ca 231 K with a hysteresis width of 10 K. Variable-temperature X-ray single-crystal diffraction revealed that the phase transition of 1 can be ascribed to coupling of pendulum-like motions of its nitro group with proton transfer in O–H···O hydrogen bonds of the 3D framework. The temperature dependence of its dielectric permittivity demonstrated a step-like change in the range of 150–280 K with remarkable dielectric anisotropy, making 1 a promising switchable dielectric material. Potential energy calculations further supported the possibility of dynamic motion of the cationic nitro group. Overall, our findings may inspire the development of other switchable dielectric phase transition materials by introducing inorganic anions into organic–inorganic hybrid systems.

Phase Transition Triggered by Ordering of Unique Pendulum-Like Motions in a Supramolecular Complex: Potassium Hydrogen Bis(dichloroacetate)-18-Crown-6

A novel supramolecular phase change material (PCM) potassium hydrogen bis(dichloroacetate)-18-crown-6 (1), which undergoes a reversible second-order phase transition at 181.8 K (Tc), has been successfully synthesized and grown as bulk crystals. DSC measurements confirm its reversible phase transition with a thermal hysteresis of 1.6 K. Dielectric permittivities also display obvious anomalies approaching Tc, being characteristic for the reversible phase transition. Variable-temperature X-ray single-crystal diffractions demonstrate that 1 behaves as a molecular rotor above room temperature in which the chlorine atoms rotate around the C–C axis of the CHCl2COO–/CHCl2COOH exhibiting distinct pendulum-like motions, and the (18-crown-6)·K+ part acts as a stator. Further studies reveal that the origin of its phase transition was ascribed to the moving close together of the whole set molecules from the original position, which is induced by the order–disorder transformation of the pendulum-like motions of the CHC...

Supramolecular Bola-Like Ferroelectric: 4-Methoxyanilinium Tetrafluoroborate-18-crown-6

Molecular motion is one of the structural foundations for the development of functional molecular materials such as artificial motors and molecular ferroelectrics. Herein, we show that pendulum-like motion of the terminal group of a molecule causes a ferroelectric phase transition. Complex 4-methoxyanilinium tetrafluoroborate-18-crown-6 ([C(7)H(10)NO(18-crown-6)](+)[BF(4)](-), 1) shows a second-order ferroelectric phase transition at 127 K, together with an abrupt dielectric anomaly, Debye-type relaxation behavior, and the symmetry breaking confirmed by temperature dependence of second harmonic generation effect. The origin of the polarization is due to the order-disorder transition of the pendulum-like motions of the terminal para-methyl group of the 4-methoxyanilinium guest cation; that is, the freezing of pendulum motion at low temperature forces significant orientational motions of the guest molecules and thus induces the formation of the ferroelectric phase. The supramolecular bola-like ferroelectric is distinct from the precedent ferroelectrics and will open a new avenue for the design of polar functional materials.

On the orbital stability of pendulum-like motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev-Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the basis of a nonlinear analysis.

Uncertainties in the measurement of the atmospheric velocity due to balloon-gondola pendulum-like motions

Balloons lead to the highest vertical resolution of air velocity data actually attainable from atmospheric soundings. However, the pendulum-like motion of the balloon-gondola system may significantly affect these measurements if the distance between balloon and gondola is large. This may prevent the study of the highest vertical resolution range obtained. Also, if not appropriately discriminated, these fluctuations could be confused with small scale or turbulent oscillations of the atmosphere. It is shown from simple energy considerations that horizontal and vertical wind velocity perturbations introduced in the observations by the pendulum motion may usually be comparable to typical measurements. Vertical velocity data that were obtained with an instrumented gondola in a zero pressure balloon, which typically reach the lower stratosphere, are analyzed and found to be in agreement with the above statements. The pendulum-like behavior in this sounding seems to be stimulated by the buoyant oscillation of the atmosphere.

Invariant sets in the Goryachev–Chaplygin problem: existence, stability and branching

The existence, stability and branching of invariant sets in the problem of the motion of a heavy rigid body with a fixed point, which satisfies the Goryachev–Chaplygin conditions, are discussed. Both trivial invariant sets, in which the pendulum-like motions of a Goryachev–Chaplygin spinning top lie, as well as non-trivial invariant sets, in which the motion of the top is described by elliptic functions of time, are investigated.

The pendulum-like motions of a rigid body in the Goryachev-Chaplygin case

The motion of a heavy rigid body with a single fixed point in a uniform gravity field is considered. The geometry of the mass of the body and the initial conditions of its motion correspond to the case of Goryachev-Chaplygin integrability [1, 2]. In this case periodic pendulum-like motions exist, corresponding to oscillations or rotations of the body around an axis of dynamic symmetry, occupying a fixed horizontal position. The problem of the orbital stability of such motions is solved. An explicit solution of the linearized equations of the perturbed motion is obtained and it is shown that, in the linear approximation, the oscillations and rotations of the body are orbitally stable, while the non-linear problem of stability is always a resonance problem: for any amplitude of the oscillations (or any angular velocity of rotation) of the body in unperturbed motion its perturbed motion is such that fourthorder resonance occurs (two non-unity multipliers are pure imaginary and equal to ± i). It is shown that, in the non-linear formulation of the problem, the pendulum-like oscillations of the body are always orbitally unstable, while the rotations are stable.

The sufficient conditions for the existence of asymptotically pendulum-like motions of a heavy rigid body with a fixed point

The present paper continues the study of the asymptotically pendulum-like motions (APM) of a heavy rigid body begun in /1/, where a specific mass distribution is not assumed here a priori. The first Lyapunov method /2/ is used to obtain new sufficient conditions for the existence of APM, which cannot be described by the well-known particular solutions of the Euler-Poisson equations.

Asymptotically pendulum-like motions of the hess-appel'rot gyroscope

Asymptotically pendulum-like motions of a heavy rigid body whose centre of gravity lies on the perpendicular to the circular cross-section of the gyration ellipsoid (the Hess-Appel'rot gyroscope) is investigated. Lyapunov's theorem is used to show that the initial position and initial angular velocity of this gyroscope can also be chosen such, that its motion will tend asymptotically, as time increases without limit, to rotation about the horizontal axis. Since in this case the initial conditions do not satisfy the invariant Hess relation, it follows that the results described cannot be obtained by direct generalisation of /1/ where the asymptotically pendulum-like motions were obtained for the special case of the Hess solution by constructing the phase trajectories. Various examples of asymptotic motions in the classical problem of the motion of a heavy rigid body with a fixed point are shown in /1–5/.

The Physiology of Training the Functional Development of the Labyrinthine Function Through the Daily Repetition of Rotary, Centrifugal, See-Saw and Pendulum-Like Motions: (Synopsis of the Dialogue of the Film, 16 mm with soundtrack in colour, 26 min.)

This film is concerned with the ‘response decline” associated with the post-rotatory nystagmus which is induced by rotation repeated daily. It differs, however, from many previous reports on this subject in that a new concept of the equilibrating function is established in this film. This concept recognizes that a hitherto unknown and new labyrinthine reflex of a higher order concerning the reflex movement of the head, neck, trunk and tail of a fowl is induced during rotation by means of repeated rotations. The establishment of this new labyrinthine reflex of a higher order is clearly evidenced not only by the repetition of rotation but also by the repetition of centrifugal, see-saw and pendulum-like motions. Therefore, repetition of the various movements mentioned has been called ‘training”.