A system of one dimensional balls with gravity

Wojtkowski, Maciej P.

doi:10.1007/bf02125698

Cited by 59 publications

(46 citation statements)

References 28 publications

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“…For θ = 45 o the motion is completely integrable, while for θ > 45 o the motion is chaotic. Wojtkowski refers to this geometry as "fat billiards" and has rigor-ously proven that they have a single, ergodic component [15]. These results have also been confirmed through experiments for an optical billiard with ultra cold atoms [16].…”

Section: Introductionmentioning

confidence: 78%

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Dynamics of a dissipative, inelastic gravitational billiard

Hartl¹,

Miller²,

Mazzoleni³

2013

Phys. Rev. E

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The seminal physical model for investigating formulations of nonlinear dynamics is the billiard. Gravitational billiards provide an experimentally accessible arena for their investigation. We present a mathematical model that captures the essential dynamics required for describing the motion of a realistic billiard for arbitrary boundaries, where we include rotational effects and additional forms of energy dissipation. Simulations of the model are applied to parabolic, wedge and hyperbolic billiards that are driven sinusoidally. The simulations demonstrate that the parabola has stable, periodic motion, while the wedge and hyperbola (at high driving frequencies) appear chaotic. The hyperbola, at low driving frequencies, behaves similarly to the parabola; i.e., has regular motion. Direct comparisons are made between the model's predictions and previously published experimental data. The representation of the coefficient of restitution employed in the model resulted in good agreement with the experimental data for all boundary shapes investigated. It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.

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Section: Introductionmentioning

confidence: 78%

“…Note that for the two-dimensional case, the following generalized speeds are zero: u 1 , u 6 and u ′ 6 . From Equations (15) and (16), we define the velocity of the point P of the sphere that comes into contact with the boundary as…”

Section: A Backgroundmentioning

confidence: 99%

Dynamics of a dissipative, inelastic gravitational billiard

Hartl¹,

Miller²,

Mazzoleni³

2013

Phys. Rev. E

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“…The system of two falling balls, introduced by Wojtkowski in [17], describes the motion of two point particles of masses m 1 and m 2 that move along the vertical half-line, subject to constant gravitational force, and collide elastically with each other and the floor. We consider the case when the lower ball is heavier (i.e.…”

Section: Setup and Notationsmentioning

confidence: 99%

“…We consider the case when the lower ball is heavier (i.e. m 1 > m 2 ), which corresponds to ergodic and hyperbolic dynamics (as shown in [12] and [17]). As the action of ball to ball collisions depends only on the ratio of the two masses we rescale these masses such that m 1 + m 2 = 1.…”

Section: Setup and Notationsmentioning

confidence: 99%

The flow of two falling balls mixes rapidly

Bálint

Varga

2016

Nonlinearity

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In this paper we study the system of two falling balls in continuous time. We modell the system by a suspension flow over a two dimensional, hyperbolic base map. By detailed analysis of the geometry of the system we identify special periodic points and show that the ratio of certain periods in continuous time is Diophantine for almost every value of the mass parameter in an interval. Using results of Melbourne ([13]) and our previous achievements [1] we conclude that for these values of the parameter the flow mixes faster than any polynomial. Even though the calculations are presented for the specific physical system, the method is quite general and can be applied to other suspension flows, too.

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“…Here, however, we do not have an infinite energy limit but instead zero distance: The angle α is also the wedge angle, which is the only system parameter, and at π/2 the wedge degenerates into a line. For α > π/4 the orbit is elliptic, at π/4 where the billiard becomes integrable it is invers parabolic and for smaller angles it is inverse hyperbolic, as must be the case because the billiard is ergodic for α < π/4 [34].…”

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confidence: 94%

Linear stability in billiards with potential

Dullin

1998

Nonlinearity

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A general formula for the linearized Poincaré map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. For the case without potential this reduces to well known formulas. Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.

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A system of one dimensional balls with gravity

Abstract: We introduce a Hamiltonian system with many degrees of freedom for which the nonvanishing of (some) Lyapunov exponents almost everywhere can be established analytically.

Cited by 59 publications

References 28 publications

Dynamics of a dissipative, inelastic gravitational billiard

Dynamics of a dissipative, inelastic gravitational billiard

The flow of two falling balls mixes rapidly

Linear stability in billiards with potential

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