Circular Billiard
نویسندگان
چکیده
We analyze the problem of perfectly elastic billiard on a circular table with exactly one permitted bounce. We present a new and intuitively appealing geometric derivation of the solution. Analyzing the solution with respect to the number of permitted paths for any given scenario, we nd an analytical expression for a separatrix between regions with two and four solutions. We identify and discuss symmetry aspects of the problem and singular points on the billiard table. Finally we apply the results to an optical experiment which can be performed in any classroom.
منابع مشابه
Two-particle circular billiards versus randomly perturbed one-particle circular billiards.
We study a two-particle circular billiard containing two finite-size circular particles that collide elastically with the billiard boundary and with each other. Such a two-particle circular billiard provides a clean example of an "intermittent" system. This billiard system behaves chaotically, but the time scale on which chaos manifests can become arbitrarily long as the sizes of the confined p...
متن کاملSurvival probability for open spherical billiards.
We study the survival probability for long times in an open spherical billiard, extending previous work on the circular billiard. We provide details of calculations regarding two billiard configurations, specifically a sphere with a circular hole and a sphere with a square hole. The constant terms of the long-time survival probability expansions have been derived analytically. Terms that vanish...
متن کاملTime-Dependent Circular Billiard
We investigate a time-dependent circular billiard with a two-frequency driving function and derive a new simplified form for the map, which is a symplectic nontwist map. Stability boundaries and reconnection thresholds are derived for fixed points and period-two vortex pairs. An island interspersal condition is derived such that neighboring island chains of the first frequency are exactly separ...
متن کاملShortest billiard trajectories ∗
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 (cl...
متن کاملAn Introduction to Dynamical Billiards
Some billiard tables in R2 contain crucial references to dynamical systems but can be analyzed with Euclidean geometry. In this expository paper, we will analyze billiard trajectories in circles, circular rings, and ellipses as well as relate their charactersitics to ergodic theory and dynamical systems.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM Review
دوره 40 شماره
صفحات -
تاریخ انتشار 1998