Ball Orbits Research Articles

Centrally symmetric analytic plane domains are spectrally determined in this class

We prove that, under some generic non-degeneracy assumptions, real analytic centrally symmetric plane domains are determined by their Dirichlet (resp. Neumann) spectra. We prove that the conditions are open-dense for real analytic strictly convex domains. One step is to use a Maslov index calculation to show that the second derivative of the defining function of a centrally symmetric domain at the endpoints of a bouncing ball orbit is a spectral invariant. This is also true for up-down symmetric domains, removing an assumption from the proof in that case.

Trace singularities in obstacle scattering and the Poisson relation for the relative trace

We consider the case of scattering by several obstacles in {mathbb {R}}^d, d ge 2 for the Laplace operator Delta with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators Delta _1 and Delta _2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator g(Delta ) - g(Delta _1) - g(Delta _2) + g(Delta _0) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function xi _mathrm {rel}(lambda ) = -frac{1}{pi } {text {Im}}(Xi (lambda )), where Xi (lambda ) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of xi _mathrm {rel}. In particular we prove that {hat{xi }}_mathrm {rel} is real-analytic near zero and we relate the decay of Xi (lambda ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function Xi (lambda ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.

The leaking soft stadium

We soften the non zero y-boundary on a Bunimovich like quarter-stadium. The smoothing procedure is performed via an exponent monomial potential, the system becomes partially reflective, preserving the particle’s translation and rotational motion. By increasing the exponent value, the stadium’s boundaries become rigid and the system’s dynamics reaches a chaotic regime. We set a leaking soft stadium family by opening a limited region located at some place of its basis’s boundary, throughout which the particles can leak out. This work is an extension of our recently reported paper on this matter. We chase the particle’s trajectory and focus on the stadium transient behavior by means of the statistical analysis of the survival probability on the marginal orbits that never leave the system, the so called bouncing ball orbits. We compare these family orbits with the billiard’s transient chaos orbits.

Humanoid Fast Tennis Swing Motion by Real-time Ball Orbit Predition and Real-time Joint Trajectory Modification

When a humanoid robot does a task that requires the whole-body fast motion such as tennis forehand swing, it is difficult for the robot to exploit full potential of the actuators’ performance and to modify the joint trajectory in real time. We propose a method of whole-body fast motion planning with balance margin. By using this method, we achieved tennis forehand stroke with a humanoid to see a ball with a stereo camera on the its head.

Survival probability for the stadium billiard

We consider the open stadium billiard, consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides. Due to the hyperbolic nature of the stadium billiard, the initial decay of trajectories, due to loss through the hole, appears exponential. However, some trajectories (bouncing ball orbits) persist and survive for long times and therefore form the main contribution to the survival probability function at long times. Using both numerical and analytical methods, we concur with previous studies that the long-time survival probability for a reasonably small hole drops like C o n s t a n t × ( t i m e ) − 1 ; here we obtain an explicit expression for the Constant.

Inverse spectral problem for analytic domains, II: ℤ2-symmetric domains

This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary, and on a Feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral. The algorithm is particularly effective for Euclidean plane domains possessing a ℤ 2 symmetry which reverses the orientation of a bouncing ball orbit. It is also very effective for domains with dihedral symmetries. For simply connected analytic Euclidean plane domains in either symmetry class, we prove that the domain is determined within the class by either its Dirichlet or Neumann spectrum. This improves and generalizes the best prior inverse result that simply connected analytic plane domains with two symmetries are spectrally determined within that class.

Dynamic analysis and grinding tracks in the magnetic fluid grinding system

Abstract To model the effects of the geometrical imperfections on the ball motion and its grinding track, it is therefore necessary to combine a dynamic model of the support system of balls with the previous model. For the geometrical imperfections on the ball, because of the interaction between the contact loads and the ball-spin speed, it causes the friction contact condition to remain at the interfaces with lower contact loads and lower ball-spin speeds in the separation case at the initial stage. Consequently, the variation in the ball-spin angle and the area covered by the grinding tracks is small. However, when the intermittent separation occurs at the geometrical imperfections on the ball orbit, it causes a large oscillation in the ball-spin angle and the ball-spin speed. Consequently, the effect of the imperfections in the ball orbit on the area covered by the grinding tracks is larger than that of the ball geometry. Ball–ball contacts cause a large oscillation in the ball-spin angle resulting in a uniform distribution of the grinding tracks. Hence, the effect of ball–ball contacts is one of the most important mechanisms in achieving a uniform distribution of the grinding tracks.

Bouncing ball orbits and symmetry breaking effects in a three-dimensional chaotic billiard

We study the classical and quantum mechanics of a three-dimensional stadium billiard. It consists of two quarter cylinders that are rotated with respect to each other by 90 degrees and it is classically chaotic. The billiard exhibits only a few families of nongeneric periodic orbits. We introduce an analytic method for their treatment. The length spectrum can be understood in terms of the nongeneric and unstable periodic orbits. For unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory. For equal radii the billiard exhibits an additional symmetry. We investigated the effects of symmetry breaking on spectral properties. Moreover, for equal radii, we observe a small deviation of the level statistics from random matrix theory. This led to the discovery of stable and marginally stable orbits, which are absent for unequal radii.

112 ボールバランサに起因する回転軸系の自励振動の特性

The ball balancer has been used for the vibration suppression method in rotor system. It has a superior characteristic that the vibration amplitude is reduced to zero by it at the rotational speed range higher than the major critical speed. However, the ball balancer may cause a self-excited vibration when the balls rotate in the disk, and this self-excited vibration results in the large amplitude vibration. In this paper, the occurrence region and vibration characteristics of the self-excited vibration are investigated. The theoretical analysis is performed and a set of the fundamental equations governing the self-excited vibration is obtained. The influences of the parameters, such as, damping of the ball motion, gyro-effect of the system, ball mass, radius of the ball orbit, and the system nonlinearity are explained. As the result, it is shown that the damping of the ball motion and the ball mass have the effect on decreasing the occurrence region of the self excited vibration.

Experimental versus numerical eigenvalues of a Bunimovich stadium billiard: a comparison.

We compare the statistical properties of eigenvalue sequences for a gamma=1 Bunimovich stadium billiard. The eigenvalues have been obtained in two ways: one set results from a measurement of the eigenfrequencies of a superconducting microwave resonator (real system), and the other set is calculated numerically (ideal system). We show influence of mechanical imperfections of the real system in the analysis of the spectral fluctuations and in the length spectra compared to the exact data of the ideal system. We also discuss the influence of a family of marginally stable orbits, the bouncing ball orbits, in two microwave stadium billiards with different geometrical dimensions.

How chaotic is the stadium billiard? A semiclassical analysis

The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. I show that this belief is erroneous. The Gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Green function. Semiclassical contributions to the trace show an $\hbar$ - dependent transition from hard chaos to integrable behavior for trajectories approaching the bouncing ball orbits. A whole region in phase space surrounding the marginal stable family acts, semiclassically, like a stable island with boundaries being explicitly $\hbar$-dependent. The localized bouncing ball states found in the billiard derive from this semiclassically stable island. The bouncing ball orbits themselves, however, do not contribute to individual eigenvalues in the spectrum. An EBK-like quantization of the regular bouncing ball eigenstates in the stadium can be derived. The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics.

Decay of classical chaotic systems: The case of the Bunimovich stadium.

The escape of an ensemble of particles from the Bunimovich stadium via a small hole has been studied numerically. The decay probability starts out exponentially but has an algebraic tail. The weight of the algebraic decay tends to zero for vanishing hole size. This behavior is explained by the slow transport of the particles close to the marginally stable bouncing ball orbits. It is contrasted with the decay function of the corresponding quantum system. \textcopyright{} 1996 The American Physical Society.

Dynamic and coupling influences on basic speed ratio of an angular contact bearing

A general expression for basic speed ratio, a fundamental characteristic number for an angular contact ball bearing, is used to explain some previously reported data. Estimates of ball-race slip induced by ball orbit drag and separately by ball spin drag are given. Some dynamic effects are also discussed.