The Geometry Of Hamilton And Lagrange Spaces
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The Geometry of Hamilton and Lagrange Spaces
Author: R. Miron
language: en
Publisher: Springer Science & Business Media
Release Date: 2001-05-31
This monograph presents for the first time the foundations of Hamilton Geometry. The concept of Hamilton Space, introduced by the first author and investigated by the authors, opens a new domain in differential geometry with large applications in mechanics, physics, optimal control, etc. The book consists of thirteen chapters. The first three chapters present the topics of the tangent bundle geometry, Finsler and Lagrange spaces. Chapters 4-7 are devoted to the construction of geometry of Hamilton spaces and the duality between these spaces and Lagrange spaces. The dual of a Finsler space is a Cartan space. Even this notion is completely new, its geometry has the same symmetry and beauty as that of Finsler spaces. Chapter 8 deals with symplectic transformations of cotangent bundle. The last five chapters present, for the first time, the geometrical theory and applications of Higher-Order Hamilton spaces. In particular, the case of order two is presented in detail. Audience: mathematicians, geometers, physicists, and mechanicians. This volume can also be recommended as a supplementary graduate text.
The Geometry of Hamilton and Lagrange Spaces
Author: R. Miron
language: en
Publisher: Springer Science & Business Media
Release Date: 2006-04-11
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.
The Geometry of Higher-Order Hamilton Spaces
Author: R. Miron
language: en
Publisher: Springer Science & Business Media
Release Date: 2003-10-31
This book is the first to present an overview of higher-orderHamilton geometry with applications to higher-order Hamiltonianmechanics. It is a direct continuation of the book "The Geometry ofHamilton and" "Lagrange Spaces," (Kluwer Academic Publishers,2001). It contains the general theory of higher order Hamilton spaces"H," "k>=1," semisprays, the canonical nonlinearconnection, the N-linear metrical connection and their structureequations, and the Riemannian almost contact metrical model of thesespaces. In addition, the volume also describes new developments suchas variational principles for higher order Hamiltonians; Hamilton-Jacobi equations; higher order energies and law ofconservation; Noether symmetries; Hamilton subspaces of order k andtheir fundamental equations. The duality, via Legendre transformation, between Hamilton spaces of order k and Lagrange spaces of the sameorder is pointed out. Also, the geometry of Cartan spaces of order k=1 is investigated in detail. This theory is useful intheconstruction of geometrical models in theoretical physics, mechanics, dynamical systems, optimal control, biology, economy etc."Audience: " Mathematicians, geometers, physicists and engineers.The volume can be recommended as a supplementary graduate text.