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E-raamat: Exterior Billiards: Systems with Impacts Outside Bounded Domains

  • Formaat: PDF+DRM
  • Ilmumisaeg: 13-Sep-2012
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781461444817
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Exterior Billiards: Systems with Impacts Outside Bounded DomainsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 13-Sep-2012
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781461444817
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A billiard is a dynamical system in which a point particle alternates between free motion and specular reflections from the boundary of a domain. Exterior Billiards presents billiards in the complement of domains and their applications in aerodynamics and geometrical optics. This book distinguishes itself from existing literature by presenting billiard dynamics outside bounded domains, including scattering, resistance, invisibility and retro-reflection. It begins with an overview of the mathematical notations used throughout the book and a brief review of the main results. Chapters 2 and 3 are focused on problems of minimal resistance and Newton’s problem in media with positive temperature. In chapters 4 and 5, scattering of billiards by nonconvex and rough domains is characterized and some related special problems of optimal mass transportation are studied. Applications in aerodynamics are addressed next and problems of invisibility and retro-reflection within the framework of geometric optics conclude the text. The book will appeal to mathematicians working in dynamical systems and calculus of variations. Specialists working in the areas of applications discussed will also find it useful.

Surveying the nature of dynamical systems, in which a point particle alternates between free motion and specular reflections from the boundary of a domain, this book assesses the ways in which they can be applied to aerodynamics as well as geometrical optics.

Arvustused

From the reviews:

The book under review is a very nice presentation of several problems related to the aerodynamics of bodies in highly rarefied media. The book is very pleasant to read; the various mathematical models are clearly presented, and even a nonspecialist in the field can follow the presentation of the problems. Several pictures that are included greatly help the reader . (Giuseppe Buttazzo, SIAM Review, Vol. 56 (1), March, 2014)

1 Notation and Synopsis of Main Results
1(20)
1.1 Definition of Resistance
1(3)
1.2 Newton's Aerodynamic Problem
4(3)
1.3 Problems of Least Resistance to Translational Motion of Nonconvex Bodies
7(1)
1.4 Generalized Newton's Problem in Media with Positive Temperature
8(3)
1.5 Scattering in Billiards
11(1)
1.6 Problems of Optimal Mass Transportation
12(3)
1.7 Optimizing the Mean Resistance
15(2)
1.8 Dynamics of a Spinning Rough Disc
17(2)
1.9 Billiards Possessing Extremal Aerodynamic Properties
19(2)
2 Problem of Minimum Resistance to Translational Motion of Bodies
21(34)
2.1 Bodies Inscribed in a Circular Cylinder
22(6)
2.1.1 The Class of Bodies with Fixed Horizontal Projection
23(3)
2.1.2 The Class of Sets Containing a Section of the Cylinder
26(2)
2.2 Bodies Inscribed in an Arbitrary Cylinder
28(2)
2.3 Bodies Modified in a Neighborhood of Their Boundary
30(13)
2.3.1 Preliminary Constructions
32(5)
2.3.2 Proof of Theorem 2.3
37(3)
2.3.3 Proof of Theorem 2.2
40(3)
2.4 Two-Dimensional Problem
43(6)
2.4.1 Minimum Resistance of Convex Bodies
43(1)
2.4.2 Minimum Resistance of Nonconvex Bodies
44(5)
2.5 Minimum Specific Resistance of Unbounded Bodies
49(6)
3 Newton's Problem in Media with Positive Temperature
55(50)
3.1 Calculation of Resistance and Statement of Minimization Problem
55(4)
3.1.1 Description of the Medium
55(2)
3.1.2 Calculation of Resistance
57(2)
3.1.3 Statement of Minimization Problem
59(1)
3.2 Auxiliary Minimization Problems
59(9)
3.2.1 Two Lemmas on the Functions p±
59(2)
3.2.2 Lemma of Reduction
61(1)
3.2.3 The Minimizing Function for d = 2
62(2)
3.2.4 The Minimizing Function for d ≥ 3
64(4)
3.3 Solution of Minimum Resistance Problem
68(9)
3.3.1 Two-Dimensional Problem
68(4)
3.3.2 The Problem in Three and More Dimensions
72(2)
3.3.3 Limiting Cases
74(3)
3.4 Gaussian Distribution of Velocities: Exact Solutions
77(5)
3.4.1 Two-Dimensional Case
78(1)
3.4.2 Three-Dimensional Case
79(3)
3.5 Proof of Auxiliary Statements
82(23)
3.5.1 Proof of Lemma 3.1
82(7)
3.5.2 Proof of Lemma 3.2
89(14)
3.5.3 Proof of Formula (3.41)
103(2)
4 Scattering in Billiards
105(46)
4.1 Scattering in the Two-Dimensional Case
108(22)
4.1.1 Measures Associated with Hollows
109(2)
4.1.2 Examples
111(4)
4.1.3 Basic Theorem
115(9)
4.1.4 Proof of Formula (4.20)
124(2)
4.1.5 Classification of Scattering Laws on Two-Dimensional Bodies
126(4)
4.2 Scattering by the Surface of Rough Bodies
130(21)
4.2.1 Proof of Theorem 4.4
135(6)
4.2.2 Proof of Theorem 4.5
141(2)
4.2.3 Proof of Lemma 4.2
143(2)
4.2.4 Proof of Lemma 4.3
145(1)
4.2.5 Resistance of Notched Arc
145(6)
5 Problems of Optimal Mass Transportation
151(32)
5.1 Statement of the One-Dimensional Problem and the Results
152(7)
5.2 Proof of Theorem 5.1
159(14)
5.3 Examples
173(4)
5.4 The Problem of Mass Transfer on a Sphere
177(6)
6 Problems on Optimization of Mean Resistance
183(14)
6.1 Two-Dimensional Case
183(9)
6.1.1 Resistance in a Medium with Temperature Zero
184(5)
6.1.2 Media with Nonzero Temperature
189(3)
6.2 Case of Higher Dimension
192(5)
7 The Magnus Effect and the Dynamics of a Rough Disc
197(22)
7.1 Description of the Effect and Statement of the Problem
198(4)
7.1.1 Statement of the Problem for a Rough Disc
198(2)
7.1.2 Summary of the Remainder of the
Chapter
200(2)
7.2 Resistance of a Rough Disc
202(6)
7.3 Magnus Effect
208(5)
7.3.1 Vector-Valued Monge-Kantorovich Problem
209(2)
7.3.2 Special Cases of Rough Discs
211(2)
7.4 Dynamics of a Rough Disc
213(3)
7.5 Conclusions and Comparison with Previous Works
216(3)
8 Invisible Bodies
219(24)
8.1 Main Constructions
220(4)
8.1.1 Definitions and Statement of Main Result
220(1)
8.1.2 Proof of Theorem 8.1
221(3)
8.2 Other Constructions of Bodies of Zero Resistance
224(4)
8.3 Properties of Bodies of Zero Resistance
228(3)
8.4 Invisibility in Several Directions
231(5)
8.4.1 Bodies Invisible in Two Directions
231(2)
8.4.2 Nonexistence of Bodies Invisible in All Directions
233(3)
8.5 Bodies Invisible from One Point
236(5)
8.6 Possible Applications of Invisible Bodies and Open Questions
241(2)
9 Retroreflectors
243(36)
9.1 Preliminaries
243(7)
9.1.1 Unbounded Bodies
244(2)
9.1.2 Basic Definitions
246(4)
9.2 Mushroom
250(1)
9.3 Tube
250(4)
9.4 Notched Angle
254(7)
9.5 Helmet
261(1)
9.6 Collection of Retroreflectors
262(2)
9.7 Proofs of Auxiliary Statements
264(15)
9.7.1 Convergence of Measures Associated with Rectangular Hollows
264(1)
9.7.2 Convergence of Measures Associated with Triangular Hollows
265(2)
9.7.3 Size of the Smallest Hollows in a Mushroom Body
267(12)
Bibliography 279(4)
Index 283
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