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1999•Chaos An Interdisciplinary Journal of Nonlinear ScienceRequires access

Dynamics of a gravitational billiard with a hyperbolic lower boundary

Matthew Ferguson, Bruce N. Miller, M. A. Thompson

Open publisher page 9 citations

Abstract

Gravitational billiards provide a simple method for the illustration of the dynamics of Hamiltonian systems. Here we examine a new billiard system with two parameters, which exhibits, in two limiting cases, the behaviors of two previously studied one-parameter systems, namely the wedge and parabolic billiard. The billiard consists of a point mass moving in two dimensions under the influence of a constant gravitational field with a hyperbolic lower boundary. An iterative mapping between successive collisions with the lower boundary is derived analytically. The behavior of the system during transformation from the wedge to the parabola is investigated for a few specific cases. It is surprising that the nature of the transformation depends strongly on the parameter values. (c) 1999 American Institute of Physics.

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What this paper is about

Gravitational billiards provide a simple method for the illustration of the dynamics of Hamiltonian systems. Here we examine a new billiard system with two parameters, which exhibits, in two limiting cases, the behaviors of two previously studied one-parameter systems, namely the wedge and parabolic billiard. The billiard consists of a point mass moving in two dimensions under the influence of a constant gravitational field with a hyperbolic lower boundary. An iterative mapping between successive collisions with the lower boundary is derived analytically. The behavior of the system during transformation from the wedge to the parabola is investigated for a few specific cases. It is surprising that the nature of the transformation depends strongly on the parameter values. (c) 1999 American Institute of Physics.

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Available abstract

Gravitational billiards provide a simple method for the illustration of the dynamics of Hamiltonian systems. Here we examine a new billiard system with two parameters, which exhibits, in two limiting cases, the behaviors of two previously studied one-parameter systems, namely the wedge and parabolic billiard. The billiard consists of a point mass moving in two dimensions under the influence of a constant gravitational field with a hyperbolic lower boundary. An iterative mapping between successive collisions with the lower boundary is derived analytically. The behavior of the system during transformation from the wedge to the parabola is investigated for a few specific cases. It is surprising that the nature of the transformation depends strongly on the parameter values. (c) 1999 American Institute of Physics.

Key concepts: Dynamical billiards, Dynamics (music), Boundary (topology), Physics, Classical mechanics, Gravitation, Mathematics, Mathematical analysis

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