An Introduction to Mathematical Billiards

دانلود کتاب An Introduction to Mathematical Billiards

Author: Utkir A Rozikov

توضیحات کتاب :

A mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets

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A mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets. The ball moves and its trajectory is defined by the ball's initial position and its initial speed vector. The ball's reflections from the boundary of the table are assumed to have the property that the reflection and incidence angles are the same. This book comprehensively presents known results on the behavior of a trajectory of a billiard ball on a planar table (having one of the following forms: circle, ellipse, triangle, rectangle, polygon and some general convex domains). It provides a systematic review of the theory of dynamical systems, with a concise presentation of billiards in elementary mathematics and simple billiards related to geometry and physics.

The description of these trajectories leads to the solution of various questions in mathematics and mechanics: problems related to liquid transfusion, lighting of mirror rooms, crushing of stones in a kidney, collisions of gas particles, etc. The analysis of billiard trajectories can involve methods of geometry, dynamical systems, and ergodic theory, as well as methods of theoretical physics and mechanics, which has applications in the fields of biology, mathematics, medicine, and physics.

ادامه ...

Ebook details:
عنوان: An Introduction to Mathematical Billiards
نویسنده: Utkir A Rozikov
ناشر: WSPC (December 7, 2018)
زبان: English
شابک: 9813276460, 978-9813276468
حجم: 10 Mb
فرمت: True Pdf

ادامه ...

Intro; Contents; Preface; Introduction; 1. Dynamical systems and mathematical billiards; 1.1 Discrete-time dynamical systems; 1.1.1 Definitions and the main problem; 1.1.2 One-dimensional systems; 1.1.3 Multi-dimensional linear systems; 1.1.4 Multi-dimensional non-linear systems; 1.2 Continuous-time dynamical systems; 1.3 Definitions and problems of billiards; 1.3.1 Definitions; 1.3.2 The billiard as a two-dimensional non-linear dynamical system; 1.3.3 The main problem; 2. Billiards in elementary mathematics; 2.1 Pouring problems; 2.2 Billiard in the circle 3.3.3 Periodic trajectories cover the triangle3.3.4 Mirror periodic trajectories; 3.3.5 Instability of periodic trajectories; 3.4 Elliptical billiard tables; 3.4.1 Reflection law of ellipse; 3.4.2 First case: Ball passes along a focus; 3.4.3 Second case: The first shot passes between the foci of the ellipse; 3.4.4 Third case: The first shot does not pass between the foci of the ellipse; 3.5 Birkhoff theorems; 3.5.1 Caustics and mirror equation; 3.5.2 n-periodic trajectories on a convex table; 3.5.3 The perimeter length function; 3.6 Billiard on a polygonal table; 3.6.1 A rectangular billiard 3.6.2 Billiard paths connecting given points3.6.3 Fagnano billiard trajectories in a convex polygon; 3.6.4 Periodic billiard trajectory in a polygon; 3.7 Chaotic billiards; 4. Billiards and physics; 4.1 Phase space; 4.2 Physics of billiards; 4.2.1 Motion and collisions of balls; 4.2.2 Fermat principle; 4.3 Mechanical interpretations of three-periodic points; 4.4 Billiard trajectories of light; 4.4.1 A construction of a trap for light; 4.4.2 Corner reflector; 4.4.3 Crushing of stones in a kidney; 4.4.4 Lighting problems of a non-convex area 4.5 The mechanical interpretation of billiard trajectories in right triangles4.6 Billiard of elementary one-dimensional elastic collisions of three particles; 4.6.1 Three particles on an infinite line; 4.6.2 Triangular billiard: Three particles on a ring; 4.7 n-particle gas; 4.8 Broken ray tomography; Bibliography; Index

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