The Geometry of Hamilton and Lagrange Spaces / Edition 1

by R. Miron, Dragos Hrimiuc, Hideo Shimada, Sorin V. Sabau

Overview

The title of this book is no surprise for people working in the field of Analytical Mechanics. However, the geometric concepts of Lagrange space and Hamilton space are completely new. The geometry of Lagrange spaces, introduced and studied in [76],[96], was ext- sively examined in the last two decades by geometers and physicists from Canada, Germany, Hungary, Italy, Japan, Romania, Russia and U.S.A. Many international conferences were devoted to debate this subject, proceedings and monographs were published [10], [18], [112], [113],... A large area of applicability of this geometry is suggested by the connections to Biology, Mechanics, and Physics and also by its general setting as a generalization of Finsler and Riemannian geometries. The concept of Hamilton space, introduced in [105], [101] was intensively studied in [63], [66], [97],... and it has been successful, as a geometric theory of the Ham- tonian function the fundamental entity in Mechanics and Physics. The classical Legendre’s duality makes possible a natural connection between Lagrange and - miltonspaces. It reveals new concepts and geometrical objects of Hamilton spaces that are dual to those which are similar in Lagrange spaces. Following this duality Cartan spaces introduced and studied in [98], [99],..., are, roughly speaking, the Legendre duals of certain Finsler spaces [98], [66], [67]. The above arguments make this monograph a continuation of [106], [113], emphasizing the Hamilton geometry.

Product Details

ISBN-13:	9780792369264
Publisher:	Springer Netherlands
Publication date:	05/31/2001
Series:	Fundamental Theories of Physics , #118
Edition description:	2002
Pages:	338
Product dimensions:	6.14(w) x 9.21(h) x 0.36(d)

Table of Contents

The geometry of tangent bundle.- Finsler spaces.- Lagrange spaces.- The geometry of cotangent bundle.- Hamilton spaces.- Cartan spaces.- The duality between Lagrange and Hamilton spaces.- Symplectic transformations of the differential geometry of T*M.- The dual bundle of a k-osculator bundle.- Linear connections on the manifold T*2M.- Generalized Hamilton spaces of order 2.- Hamilton spaces of order 2.- Cartan spaces of order 2.

Editorial Reviews

For those working in the field of analytical mechanics, the geometric concepts of Lagrange space and Hamilton space are new. Hamilton space as a geometric theory of the Hamiltonian function, introduced by Miron (Al. I. Cuza U., Iasi, Romania) and investigated with his coauthors, opens a new domain in differential geometry with applications to problems of electromagnetism, gauge theory, and relativity in mechanics, optimal control, and physics. Prior to elaborating on the natural connection between Lagrange and Hamilton spaces, the authors overview the geometry of the tangent bundle and Finsler spaces. Then they discuss the cotangent bundle T*M and Hamilton space of higher order. Includes 178 references. This monograph is suitable as a supplementary graduate text. Annotation c. Book News, Inc., Portland, OR (booknews.com)