共查询到20条相似文献,搜索用时 15 毫秒
1.
V. V. Kozlov 《Proceedings of the Steklov Institute of Mathematics》2011,273(1):196-213
A linearized problem of stability of simple periodic motions with elastic reflections is considered: a particle moves along
a straight-line segment that is orthogonal to the boundary of a billiard at its endpoints. In this problem issues from mechanics
(variational principles), linear algebra (spectral properties of products of symmetric operators), and geometry (focal points,
caustics, etc.) are naturally intertwined. Multidimensional variants of Hill’s formula, which relates the dynamic and geometric
properties of a periodic trajectory, are discussed. Stability conditions are expressed in terms of the geometric properties
of the boundary of a billiard. In particular, it turns out that a nondegenerate two-link trajectory of maximum length is always
unstable. The degree of instability (the number of multipliers outside the unit disk) is estimated. The estimates are expressed
in terms of the geometry of the caustic and the Morse indices of the length function of this trajectory. 相似文献
2.
A. P. Markeev 《Proceedings of the Steklov Institute of Mathematics》2016,295(1):190-201
The inertial motion of a material point is analyzed in a plane domain bounded by two curves that are coaxial segments of an ellipse. The collisions of the point with the boundary curves are assumed to be absolutely elastic. There exists a periodic motion of the point that is described by a two-link trajectory lying on a straight line segment passed twice within the period. This segment is orthogonal to both boundary curves at its endpoints. The nonlinear problem of stability of this trajectory is analyzed. The stability and instability conditions are obtained for almost all values of two dimensionless parameters of the problem. 相似文献
3.
H. R. Poghosyan H. M. Babujian G. K. Savvidy 《Theoretical and Mathematical Physics》2018,197(2):1592-1610
The hyperbolic Anosov C-systems have an exponential instability of their trajectories and as such represent the most natural chaotic dynamical systems. The C-systems defined on compact surfaces of the Lobachevsky plane of constant negative curvature are especially interesting. An example of such a system was introduced in a brilliant article published in 1924 by the mathematician Emil Artin. The dynamical system is defined on the fundamental region of the Lobachevsky plane, which is obtained by identifying points congruent with respect to the modular group, the discrete subgroup of the Lobachevsky plane isometries. The fundamental region in this case is a hyperbolic triangle. The geodesic trajectories of the non-Euclidean billiard are bounded to propagate on the fundamental hyperbolic triangle. Here, we present Artin’s results, calculate the correlation functions/observables defined on the phase space of the Artin billiard, and show that the correlation functions decay exponentially with time. We use the Artin symbolic dynamics, differential geometry, and the group theory methods of Gelfand and Fomin. 相似文献
4.
In this paper we prove that any convex body of the d-dimensional Euclidean space (d ≥ 2) possesses at least one shortest generalized billiard trajectory moreover, any of its shortest generalized billiard trajectories
is of period at most d + 1. Actually, in the Euclidean plane we improve this theorem as follows. A disk-polygon with parameter r > 0 is simply the intersection of finitely many (closed) circular disks of radii r, called generating disks, having some interior point in common in the Euclidean plane. Also, we say that a disk-polygon with
parameter r > 0 is a fat disk-polygon if the pairwise distances between the centers of its generating disks are at most r. We prove that any of the shortest generalized billiard trajectories of an arbitrary fat disk-polygon is a 2-periodic one.
Also, we give a proof of the analogue result for ε-rounded disk-polygons obtained from fat disk-polygons by rounding them off using circular disks of radii ε > 0. Our theorems give partial answers to the very recent question raised by S. Zelditch on characterizing convex bodies
whose shortest periodic billiard trajectories are of period 2.
K. Bezdek partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. 相似文献
5.
Pier Vittorio Ceccherini 《Journal of Geometry》1996,57(1-2):3-8
It is known that euclidean or hyperbolic spaces are characterized among certain metric spaces by the property of linearity of the equidistant locus of pairs of points. In this paper, this linearity requirement is replaced by the requirement of convexity of the set of points which are metrically pythagorean orthogonal to a given segment at a given point. As a result a new characterization of real inner product spaces among complete, convex, externally convex metric spaces is obtained. 相似文献
6.
Roman N. Karasev 《Geometric And Functional Analysis》2009,19(2):423-428
We consider billiard trajectories in a smooth convex body in
\mathbbRd{\mathbb{R}^{d}} and estimate the number of distinct periodic trajectories that make exactly p reflections per period at the boundary of the body. In the case of prime p we obtain the lower bound (d – 2)(p – 1) + 2, which is much better than the previous estimates. 相似文献
7.
Zhongxin Zhang 《Journal of Mathematical Analysis and Applications》2010,361(1):96-107
This paper is devoted to a general similarity boundary layer equation for power-law fluids, which includes many important similarity boundary layer problems such as the Falker-Skan equation and the magnetohydrodynamic boundary layer equation which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluids along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. By a rigorous mathematical analysis, the uniqueness, existence and nonexistence results for convex solutions, normal convex solutions and generalized convex solutions to the general similarity boundary layer equation are established. Also the asymptotic behavior of the normal convex solutions at the infinity are displayed. 相似文献
8.
Hong-Kun Zhang 《应用数学学报(英文版)》2011,27(3):381-392
Bunimovich billiards are ergodic and mixing. However, if the billiard table contains very large arcs on its boundary then if there exist trajectories experience infinitely many collisions in the vicinity of periodic trajectories on the large arc. The hyperbolicity is nonuniform and the mixing rate is very slow. The corresponding dynamics are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. The study of mixing rates in intermittent chaotic systems is mo... 相似文献
9.
A. V. Arutyunov D. Yu. Karamzin F. L. Pereira 《Proceedings of the Steklov Institute of Mathematics》2016,295(1):27-32
We study the Birkhoff billiard in a convex domain with a smooth boundary γ. We show that if this dynamical system has an integral which is polynomial in velocities of degree 4 and is independent with the velocity norm, then γ is an ellipse. 相似文献
10.
F. S. Duzhin 《Journal of Mathematical Sciences》2006,138(3):5691-5698
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
closed billiard trajectories of length p. Bibliography: 7 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 113–126. 相似文献
11.
The authors consider the billiard system with finitely many convex scatters with smooth boundary satisfying the visibility assumption on the plane and prove that the closed orbits for the billiard flow is uniformly distributed. 相似文献
12.
Jian Cheng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,55(3):400-419
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. 相似文献
13.
《European Journal of Operational Research》2001,135(3):514-526
In this paper, we consider a collision detection problem that frequently arises in the field of robotics. Given a set of bodies with their initial positions and trajectories, we wish to identify the first collision that occurs between any two bodies, or to determine that none exists. For the case of bodies having linear trajectories, we construct a convex hull representation of the integer programming model of S.Z. Selim and H.A. Almohamad [European Journal of Operational Research 119 (1) (1999) 121–129], and compare the relative effectiveness in solving this problem via the resultant linear program. We also extend this analysis to model a situation in which bodies move along piecewise linear trajectories, possibly rotating at the end of each linear segment. For this case, we again compare an integer programming approach with its linear programming convex hull representation, and exhibit the effectiveness of solving a sequence of mathematical programs for each time segment over a global programming scheme which considers all segments at once. We provide computational results to illustrate the effect of various numbers of bodies present in the collision scenarios, as well as the times at which the first collision occurs. 相似文献
14.
We give a lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface
in 3-space: for odd n, there are at least 2(n-1) such trajectories. Convex plane billiards were studied by G. Birkhoff, and the case of higher dimensional billiards is considered
in our previous papers. We apply a topological approach based on the calculation of cohomology of certain configuration spaces
of points on 2-sphere.
Received: 11 June 2001 / Revised version: 26 February 2002 相似文献
15.
In this note, we consider billiards with full families of periodic orbits. It is shown that the construction of a convex billiard with a “rational” caustic (i.e., carrying only periodic orbits) can be reformulated as a problem of finding a closed curve tangent to an (N - 1)-dimensional distribution on a (2N - 1)-dimensional manifold. We describe the properties of this distribution, as well as some important consequences for billiards with rational caustics. A very particular application of our construction states that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist. Bibliography: 13 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 56–64. 相似文献
16.
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. 相似文献
17.
It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to
rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant.?Another
embedding theorem states that any -hyperbolic metric space embeds isometrically into a complete geodesic -hyperbolic space.?The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications
is a characterization of the hyperbolic plane up to rough quasi-isometries.
Submitted: October 1998, Revised version: January 1999. 相似文献
18.
Yi-Chiuan Chen 《Advances in Mathematics》2010,224(2):432-460
We present in this paper an approach to studying the topological entropy of a class of billiard systems. In this class, any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers. Combining the concept of anti-integrable limit with the theory of Lyusternik-Shnirel'man, we show that a billiard system in this class generically admits a set of non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The anti-integrable limit is the singular limit when scatterers shrink to points. In order to get around the singular limit and so as to apply the implicit function theorem, on auxiliary circles encircling these scatterers we define a length functional whose critical points are well-defined at the anti-integrable limit and give rise to billiard orbits when the scatterers are not points. Consequently, we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small. 相似文献
19.
Yadong Wu 《数学研究通讯:英文版》2015,31(1):62-70
Considering the hyperbolic affine sphere equation in a smooth strictly convex bounded domain with zero boundary values, the sharp derivative estimates of any order for its convex solution are obtained. 相似文献
20.
Sergey V. Bolotin 《Proceedings of the Steklov Institute of Mathematics》2016,295(1):45-62
In an ordinary billiard system, trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than 1, we say that the billiard is degenerate. We study those trajectories of degenerate billiards that have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as small-mass limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems that shadow the trajectories of the corresponding degenerate billiards. The proofs are based on a version of the method of an anti-integrable limit. 相似文献