Exterior Billiards [electronic resource] : Systems with Impacts Outside Bounded Domains /
Plakhov, Alexander.author.
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.
Description based upon print version of record.
Exterior Billiards; Introduction; Preface; Contents; Chapter1 Notation and Synopsis of Main Results; 1.1 Definition of Resistance; 1.2 Newton's Aerodynamic Problem; 1.3 Problems of Least Resistance to Translational Motion of Nonconvex Bodies; 1.4 Generalized Newton's Problem in Media with PositiveTemperature; 1.5 Scattering in Billiards; 1.6 Problems of Optimal Mass Transportation; 1.7 Optimizing the Mean Resistance; 1.8 Dynamics of a Spinning Rough Disc; 1.9 Billiards Possessing Extremal Aerodynamic Properties; Chapter2 Problem of Minimum Resistance to Translational Motion of Bodies
2.1 Bodies Inscribed in a Circular Cylinder2.1.1 The Class of Bodies with Fixed Horizontal Projection; 2.1.2 The Class of Sets Containing a Section of the Cylinder; 2.2 Bodies Inscribed in an Arbitrary Cylinder; 2.3 Bodies Modified in a Neighborhood of Their Boundary; 2.3.1 Preliminary Constructions; 2.3.2 Proof of Theorem 2.3; 2.3.3 Proof of Theorem 2.2; 2.4 Two-Dimensional Problem; 2.4.1 Minimum Resistance of Convex Bodies; 2.4.2 Minimum Resistance of Nonconvex Bodies; 2.5 Minimum Specific Resistance of Unbounded Bodies; Chapter3 Newton's Problem in Media with Positive Temperature
3.1 Calculation of Resistance and Statement of Minimization Problem3.1.1 Description of the Medium; 3.1.2 Calculation of Resistance; 3.1.3 Statement of Minimization Problem; 3.2 Auxiliary Minimization Problems; 3.2.1 Two Lemmas on the Functions p; 3.2.2 Lemma of Reduction; 3.2.3 The Minimizing Function for d=2; 3.2.4 The Minimizing Function for d 3; 3.3 Solution of Minimum Resistance Problem; 3.3.1 Two-Dimensional Problem; 3.3.2 The Problem in Three and More Dimensions; 3.3.3 Limiting Cases; 3.4 Gaussian Distribution of Velocities: Exact Solutions; 3.4.1 Two-Dimensional Case
3.4.2 Three-Dimensional Case3.5 Proof of Auxiliary Statements; 3.5.1 Proof of Lemma 3.1; 3.5.2 Proof of Lemma 3.2; 3.5.3 Proof of Formula (3.41); Chapter4 Scattering in Billiards; 4.1 Scattering in the Two-Dimensional Case; 4.1.1 Measures Associated with Hollows; 4.1.2 Examples; 4.1.3 Basic Theorem; 4.1.4 Proof of Formula (4.20); 4.1.5 Classification of Scattering Lawson Two-Dimensional Bodies; 4.2 Scattering by the Surface of Rough Bodies; 4.2.1 Proof of Theorem 4.4; 4.2.2 Proof of Theorem 4.5; 4.2.3 Proof of Lemma 4.2; 4.2.4 Proof of Lemma 4.3; 4.2.5 Resistance of Notched Arc
Chapter 5 Problems of Optimal Mass  Transportation5.1 Statement of the One-Dimensional Problem and the Results; 5.2 Proof of Theorem 5.1; 5.3 Examples; 5.4 The Problem of Mass Transfer on a Sphere; Chapter6 Problems on Optimization of Mean Resistance; 6.1 Two-Dimensional Case; 6.1.1 Resistance in a Medium with Temperature Zero; 6.1.2 Media with Nonzero Temperature; 6.2 Case of Higher Dimension; Chapter 7 The Magnus Effect and the Dynamics of a Rough Disc; 7.1 Description of the Effect and Statement of the Problem; 7.1.1 Statement of the Problem for a Rough Disc
7.1.2 Summary of the Remainder of the Chapter
Includes bibliographical references and index.
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.
Dynamics.
Ergodic theory.
Calculus of variations.
Mathematical models.
Dynamical Systems and Ergodic Theory.
Calculus of Variations and Optimal Control; Optimization.
Mathematical Modeling and Industrial Mathematics.