Calculus and Mechanics on Two-Point Homogenous Riemannian Spaces

ISBN-13: 9783642071270 / Angielski / Miękka / 2010 / 242 str.

Alexey V. Shchepetilov

Opis

Mathematics develops both due to demands of other sciences and due to its internal logic. The latter fact explains the attention of mathematicians to many problems, which are in close connection with basic mathematical notions, even if these problems have no direct practical applications. It is well known that the space of constant curvature is one of the basic geometry notion 208], which induced the wide ?eld for investigations. As a result there were found numerous connections of constant curvature spaces with other branches of mathematics, for example, with integrable partial dif- 1 ferential equations 36, 153, 189] and with integrable Hamiltonian systems 141]. Geodesic ?ows on compact surfaces of a constant negative curvature (with the genus 2) generate many test examples for ergodic theory (see also 3 183] and the bibliography therein). The hyperbolic space H (R) is the space of velocities in special relativity (see Sect. 7.4.1) and also arises as space-like sections in some models of general relativity.

Recenzje

From the reviews:

"This book has eight chapters and a bibliography list containing 215 references. It is written in a clear and straightforward way that makes it useful even for nonspecialists in the field. ... In particular, the book contains interesting discussions of applications of the Poincaré section method to some problems in constant curvature spaces. ... The book is a valuable complete source for many-body problems on two-point homogeneous spaces." (Alexei Tsygvintsev, Mathematical Reviews, Issue 2008 f)

Spis treści

Two-Point Homogeneous Riemannian Spaces.- Differential Operators on Smooth Manifolds.- Algebras of Invariant Differential Operators on Unit Sphere Bundles Over Two-Point Homogeneous Riemannian Spaces.- Hamiltonian Systems with Symmetry and Their Reduction.- Two-Body Hamiltonian on Two-Point Homogeneous Spaces.- Particle in a Central Field on Two-Point Homogeneous Spaces.- Classical Two-Body Problem on Two-Point Homogeneous Riemannian Spaces.- Quasi-Exactly Solvability of the Quantum Mechanical Two-Body Problem on Spheres.

Blurb

The present monograph gives a short and concise introduction to classical and quantum mechanics on two-point homogenous Riemannian spaces, with empahsis on spaces with constant curvature. Chapter 1-4 provide the basic notations from differential geometry for studying two-body dynamics in these spaces. Chapter 5 deals with the problem of finding explicitly invariant expressions for the two-body quantum Hamiltonian. Chapter 6 addresses one-body problems in a central potential. Chapter 7 studies the classical counterpart of the quantum system of chapter 5. Chapter 8 investigates some applications in the quantum realm, namely for the coulomb and oscillator potentials.