Geometrical origin of chaoticity in the bouncing ball billiard

We present a study of the chaotic behaviour of the bouncing ball billiard. The work is realised on the purpose of finding at least certain causes of separation of the neighbouring trajectories. Having in view the geometrical construction of the system, we report a clear origin of chaoticity of the bouncing ball billiard. By this we claim that in case when the floor is made of arc of circles - in a certain interval of frequencies - one can give semi-analytical estimates on chaotic behaviour.     

Variational approach to homoclinic orbits in twist maps and an application to billiard systems

We study in this article the topological entropy of billiard systems on a convex domain of the Euclidean plane. We restrict our attention to those systems whose boundary curve has positive curvature and show that for generic billiard ball systems satisfying this condition the topological entropy is positive.     

A Nonholonomic Model of the Paul Trap

In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincaré map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.     

Billiard and the five-gap theorem

Let us consider the interval [0,1) as a billiard table rectangle with perimeter 1 and a sequence F(m)∈[0,1),m∈N∪{0}, of successive rebounds of a billiard ball against the sides of a billiard rectangle. We prove that if I is an open segment of a billiard rectangle, then the differences between the successive values of m for which the F(m) lies in I, take at most one even and at most four distinct odd values.     

Lower estimates for the number of closed trajectories of generalized billiards

The mathematical study of periodic billiard trajectories is a classical question that goes back to George Birkhoff. A billiard is the motion of a particle in the absence of field of force. Trajectories of such a particle are geodesics. A billiard ball rebounds from the boundary of a given domain making the angle of incidence equal the angle of reflection. Let k be a fixed integer. Birkhoff proved a lower estimate for the number of closed billiard trajectories of length k in an arbitrary plane domain. We give a general definition of a closed billiard trajectory when the billiard ball rebounds from a submanifold of a Euclidean space. Besides, we show how in this case one can apply the Morse inequalities using the natural symmetry (a closed polygon may be considered starting at any of its vertices and with the reversed direction). Finally, we prove the following estimate. Let M be a smooth closed m-dimensional submanifold of a Euclidean space, and let p > 2 be a prime integer. Then M has at least

Unbounded Orbits for Semicircular Outer Billiard

We show that the outer billiard around a semicircle has an open ball escaping to infinity. Submitted: April 19, 2008.; Accepted: January 11, 2009.     

Building a local conceptual framework for epistemic actions in a modelling environment with experiments

A local conceptual framework for the construction of mathematical knowledge in learning environments with experiments is developed. For this purpose, the mathematical modelling framework and the epistemic action model for abstraction in context are used simultaneously. In a case study, experiments of pre-service teachers with the motion of a ball on a circular billiard table are analysed within the local conceptual framework. The role of the experiments for epistemic actions of mathematical abstractions is described. In the case study, two different types of students’ approaches to the role of experiments can be distinguished.     

Integrable discretization and deformation of the nonholonomic Chaplygin ball

The rolling of a dynamically balanced ball on a horizontal rough table without slipping was described by Chaplygin using Abel quadratures. We discuss integrable discretizations and deformations of this nonholonomic system using the same Abel quadratures. As a by-product one gets a new geodesic flow on the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.     

Rolling of a ball without spinning on a plane: the absence of an invariant measure in a system with a complete set of integrals

In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.     

The rolling ball problem on the plane revisited

By a sequence of rollings without slipping or twisting along segments of a straight line of the plane, a spherical ball of unit radius has to be transferred from an initial state to an arbitrary final state taking into account the orientation of the ball. We provide a new proof that with at most 3 moves, we can go from a given initial state to an arbitrary final state. The first proof of this result is due to Hammersley ( 1983). His proof is more algebraic than ours which is more geometric. We also showed that “generically” no one of the three moves, in any elimination of the spin discrepancy, may have length equal to an integral multiple of 2π.     

Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.     

Invariants of Isospectral Deformations and Spectral Rigidity

We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace–Beltrami operator on a compact Riemannian manifold with Robin boundary conditions. Given a Kronecker invariant torus Λ of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on Λ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with Λ. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. These invariants are obtained from the first Birkhoff invariant of the microlocal monodromy operator associated to Λ. As an application we prove spectral rigidity of the Robin boundary conditions in the case of Liouville billiard tables of dimension two in the presence of a (?/2?)² group of symmetries.     

The rolling motion of a truncated ball without slipping and spinning on a plane

This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.     

Comments on the paper by A. V. Borisov,A. A. Kilin,I. S. Mamaev “How to control the Chaplygin ball using rotors. II”

In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.     

The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside

In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler — Jacobi — Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.     

Rolling of a homogeneous ball over a dynamically asymmetric sphere

We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of “clandestine” linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.     

On the covering radius of lattice zonotopes and its relation to view-obstructions and the lonely runner conjecture

The goal of this paper is twofold; first, we show the equivalence between certain problems in geometry, such as view-obstructions, billiard ball motions, and the estimation of covering radii of lattice zonotopes. Second, we utilize the latter interpretation and provide upper bounds of said radii by virtue of the Flatness Theorem. Our results allow us to specify how rational dependencies in the view-direction influence the obstruction parameter. These problems are similar in nature to the famous Lonely Runner Problem for which we draw analogous conclusions.     

Periodic billiard paths in right triangles are unstable

A stable periodic billiard path in a triangle is a billiard path which persists under small perturbations of the triangle. This article gives a geometric proof that no right triangles have stable periodic billiard paths.      

Two-link billiard trajectories: extremal properties and stability

Two-link periodic trajectories of a plane convex billiard, when a point mass moves along a segment which is orthogonal to the boundary of the billiard at its end points, are considered. It is established that, if the caustic of the boundary lies within the billiard, then, in a typical situation, there is an even number of two-link trajectories and half of them are hyperbolic (and, consequently, unstable) and the other half are of elliptic type. An example is given of a billiard for which the caustic intersects the boundary and all of the two-link trajectories are hyperbolic. The analysis of the stability is based on an analysis of the extremum of a function of the length of a segment of a convex billiard which is orthogonal to the boundary at one of its ends.     

Shortest periodic billiard trajectories in convex bodies

We show that the length of any periodic billiard trajectory in any convex body is always at least 4 times the inradius of K; the equality holds precisely when the width of K is twice its inradius, e.g., K is centrally symmetric, in which case we prove that the shortest periodic trajectories are all bouncing ball (2-link) orbits.