Calculus and mechanics on two-point homogeneous Riemannian spaces
type: Item
instance of: scholarly article
title: Calculus and mechanics on two-point homogeneous Riemannian spaces
zbMATH Open document ID: 1115.53017
author: Alexey V. Shchepetilov
published in: Lecture Notes in Physics
publication date: 2007-01-18T00:00:00Z
review text: This monograph gives a short and concise introduction to classical and quantum mechanics on two-point homogeneous Riemannian spaces, with emphasis on spaces with constant curvature. The first four chapters provide the basic notations from differential geometry for studying two-body dynamics in these spaces. The natural spaces for the two-body problem are two-point homogeneous Riemannian spaces, because the distance between bodies in these spaces is the only invariant of their location with respect to the isometry group. In Chapter 1 the classification of these spaces is given. There are also models of compact two-point homogeneous spaces as submanifolds of Euclidean spaces or its factor spaces, different models of real hyperbolic spaces \({\mathbf H}(\mathbb R)\), \(n\geq 2\) and the description of the transition from compact to noncompact two-point homogeneous spaces in terms of the corresponding Lie algebras. Chapter 2 provides the necessary data on differential operators: basic notions of the theory of invariant differential operators on smooth manifolds, the expression of the Laplace-Beltrami operator, self-adjointness of differential operators (e.g. Schrödinger operators), the general scheme of the quantum-mechanical reduction. Chapter 3 deals with algebras \(\text{Diff}_I(Q_S)\) of invariant differential operators on unit sphere bundles \(Q_S\) over two-point homogeneous spaces \(Q\). The description of these algebras in terms of generators and relations are found, also some elements from centers of these algebras. Chapter 4 contains basic facts concerning Hamiltonian dynamical systems with symmetry and the correspondence between classical and quantum-mechanical systems. A special symplectomorphism between a reduced cotangent bundle of a homogeneous manifold and a certain factor space of a coadjoint orbit of a corresponding Lie group is constructed. Chapter 5 deals with the problem of finding an explicitly invariant expression for the two-body Hamiltonian on t…
Mathematics Subject Classification ID: 53Cxx
Mathematics Subject Classification ID: 70Hxx
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 70-02
Mathematics Subject Classification ID: 81-02
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 58E50
Mathematics Subject Classification ID: 70F99
Mathematics Subject Classification ID: 70G45
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 81Q99
zbMATH DE Number: 5116431
zbMATH Keywords: two-point homogeneous Riemannian spaces
zbMATH Keywords: two-body problem
zbMATH Keywords: differential operators
zbMATH Keywords: Hamiltonian dynamical systems
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/b11771456
OpenAlex ID: W1636777865
DOI: 10.1007/B11771456
description: scientific article; zbMATH DE number 5116431