The Geometry Of Higher Order Lagrange Spaces

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The Geometry Of Higher Order Lagrange Spaces

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Author : R. Miron
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-11

The Geometry Of Higher Order Lagrange Spaces written by R. Miron and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Mathematics categories.

This monograph is devoted to the problem of the geometrizing of Lagrangians which depend on higher-order accelerations. It presents a construction of the geometry of the total space of the bundle of the accelerations of order k>=1. A geometrical study of the notion of the higher-order Lagrange space is conducted, and the old problem of prolongation of Riemannian spaces to k-osculator manifolds is solved. Also, the geometrical ground for variational calculus on the integral of actions involving higher-order Lagrangians is dealt with. Applications to higher-order analytical mechanics and theoretical physics are included as well. Audience: This volume will be of interest to scientists whose work involves differential geometry, mechanics of particles and systems, calculus of variation and optimal control, optimization, optics, electromagnetic theory, and biology.

The Geometry Of Higher Order Hamilton Spaces

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Author : R. Miron
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

The Geometry Of Higher Order Hamilton Spaces written by R. Miron and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

This book is the first to present an overview of higher-order Hamilton geometry with applications to higher-order Hamiltonian mechanics. It is a direct continuation of the book The Geometry of Hamilton and Lagrange Spaces, (Kluwer Academic Publishers, 2001). It contains the general theory of higher order Hamilton spaces H(k)n, k>=1, semisprays, the canonical nonlinear connection, the N-linear metrical connection and their structure equations, and the Riemannian almost contact metrical model of these spaces. In addition, the volume also describes new developments such as variational principles for higher order Hamiltonians; Hamilton-Jacobi equations; higher order energies and law of conservation; Noether symmetries; Hamilton subspaces of order k and their fundamental equations. The duality, via Legendre transformation, between Hamilton spaces of order k and Lagrange spaces of the same order is pointed out. Also, the geometry of Cartan spaces of order k =1 is investigated in detail. This theory is useful in the construction of geometrical models in theoretical physics, mechanics, dynamical systems, optimal control, biology, economy etc.

The Geometry Of Hamilton And Lagrange Spaces

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Author : R. Miron
language : en
Publisher: Springer Science & Business Media
Release Date : 2001-11-30

The Geometry Of Hamilton And Lagrange Spaces written by R. Miron and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-11-30 with Mathematics categories.

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.

Jet Single Time Lagrange Geometry And Its Applications

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Author : Vladimir Balan
language : en
Publisher: John Wiley & Sons
Release Date : 2011-11-22

Jet Single Time Lagrange Geometry And Its Applications written by Vladimir Balan and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-11-22 with Mathematics categories.

Develops the theory of jet single-time Lagrange geometry and presents modern-day applications Jet Single-Time Lagrange Geometry and Its Applications guides readers through the advantages of jet single-time Lagrange geometry for geometrical modeling. With comprehensive chapters that outline topics ranging in complexity from basic to advanced, the book explores current and emerging applications across a broad range of fields, including mathematics, theoretical and atmospheric physics, economics, and theoretical biology. The authors begin by presenting basic theoretical concepts that serve as the foundation for understanding how and why the discussed theory works. Subusequent chapters compare the geometrical and physical aspects of jet relativistic time-dependent Lagrange geometry to the classical time-dependent Lagrange geometry. A collection of jet geometrical objects are also examined such as d-tensors, relativistic time-dependent semisprays, harmonic curves, and nonlinear connections. Numerous applications, including the gravitational theory developed by both the Berwald-Moór metric and the Chernov metric, are also presented. Throughout the book, the authors offer numerous examples that illustrate how the theory is put into practice, and they also present numerous applications in which the solutions of first-order ordinary differential equation systems are regarded as harmonic curves on 1-jet spaces. In addition, numerous opportunities are provided for readers to gain skill in applying jet single-time Lagrange geometry to solve a wide range of problems. Extensively classroom-tested to ensure an accessible presentation, Jet Single-Time Lagrange Geometry and Its Applications is an excellent book for courses on differential geometry, relativity theory, and mathematical models at the graduate level. The book also serves as an excellent reference for researchers, professionals, and academics in physics, biology, mathematics, and economics who would like to learn more about model-providing geometric structures.

The Geometry Of Lagrange Spaces Theory And Applications

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Author : R. Miron
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

The Geometry Of Lagrange Spaces Theory And Applications written by R. Miron and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Science categories.

Differential-geometric methods are gaining increasing importance in the understanding of a wide range of fundamental natural phenomena. Very often, the starting point for such studies is a variational problem formulated for a convenient Lagrangian. From a formal point of view, a Lagrangian is a smooth real function defined on the total space of the tangent bundle to a manifold satisfying some regularity conditions. The main purpose of this book is to present: (a) an extensive discussion of the geometry of the total space of a vector bundle; (b) a detailed exposition of Lagrange geometry; and (c) a description of the most important applications. New methods are described for construction geometrical models for applications. The various chapters consider topics such as fibre and vector bundles, the Einstein equations, generalized Einstein--Yang--Mills equations, the geometry of the total space of a tangent bundle, Finsler and Lagrange spaces, relativistic geometrical optics, and the geometry of time-dependent Lagrangians. Prerequisites for using the book are a good foundation in general manifold theory and a general background in geometrical models in physics. For mathematical physicists and applied mathematicians interested in the theory and applications of differential-geometric methods.

The Geometry Of Ordinary Variational Equations

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Author : Olga Krupkova
language : en
Publisher: Springer
Release Date : 2006-11-14

The Geometry Of Ordinary Variational Equations written by Olga Krupkova and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.

The book provides a comprehensive theory of ODE which come as Euler-Lagrange equations from generally higher-order Lagrangians. Emphasis is laid on applying methods from differential geometry (fibered manifolds and their jet-prolongations) and global analysis (distributions and exterior differential systems). Lagrangian and Hamiltonian dynamics, Hamilton-Jacobi theory, etc., for any Lagrangian system of any order are presented. The key idea - to build up these theories as related with the class of equivalent Lagrangians - distinguishes this book from other texts on higher-order mechanics. The reader should be familiar with elements of differential geometry, global analysis and the calculus of variations.

The Geometry Of Hamilton And Lagrange Spaces

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Author : R. Miron
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-11

The Geometry Of Hamilton And Lagrange Spaces written by R. Miron and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-04-11 with Mathematics categories.

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.

Lagrange And Finsler Geometry

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Author : P.L. Antonelli
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09

Lagrange And Finsler Geometry written by P.L. Antonelli and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.

The differential geometry of a regular Lagrangian is more involved than that of classical kinetic energy and consequently is far from being Riemannian. Nevertheless, such geometries are playing an increasingly important role in a wide variety of problems in fields ranging from relativistic optics to ecology. The present collection of papers will serve to bring the reader up-to-date on the most recent advances. Subjects treated include higher order Lagrange geometry, the recent theory of -Lagrange manifolds, electromagnetic theory and neurophysiology. Audience: This book is recommended as a (supplementary) text in graduate courses in differential geometry and its applications, and will also be of interest to physicists and mathematical biologists.

Handbook Of Differential Geometry

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Author : Franki J.E. Dillen
language : en
Publisher: Elsevier
Release Date : 2005-11-29

Handbook Of Differential Geometry written by Franki J.E. Dillen and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-11-29 with Mathematics categories.

In the series of volumes which together will constitute the "Handbook of Differential Geometry" we try to give a rather complete survey of the field of differential geometry. The different chapters will both deal with the basic material of differential geometry and with research results (old and recent). All chapters are written by experts in the area and contain a large bibliography. In this second volume a wide range of areas in the very broad field of differential geometry is discussed, as there are Riemannian geometry, Lorentzian geometry, Finsler geometry, symplectic geometry, contact geometry, complex geometry, Lagrange geometry and the geometry of foliations. Although this does not cover the whole of differential geometry, the reader will be provided with an overview of some its most important areas. . Written by experts and covering recent research . Extensive bibliography . Dealing with a diverse range of areas . Starting from the basics

Introduction To Global Variational Geometry

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Author : Demeter Krupka
language : en
Publisher: Springer
Release Date : 2015-01-13

Introduction To Global Variational Geometry written by Demeter Krupka and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-01-13 with Mathematics categories.

The book is devoted to recent research in the global variational theory on smooth manifolds. Its main objective is an extension of the classical variational calculus on Euclidean spaces to (topologically nontrivial) finite-dimensional smooth manifolds; to this purpose the methods of global analysis of differential forms are used. Emphasis is placed on the foundations of the theory of variational functionals on fibered manifolds - relevant geometric structures for variational principles in geometry, physical field theory and higher-order fibered mechanics. The book chapters include: - foundations of jet bundles and analysis of differential forms and vector fields on jet bundles, - the theory of higher-order integral variational functionals for sections of a fibred space, the (global) first variational formula in infinitesimal and integral forms- extremal conditions and the discussion of Noether symmetries and generalizations,- the inverse problems of the calculus of variations of Helmholtz type- variational sequence theory and its consequences for the global inverse problem (cohomology conditions)- examples of variational functionals of mathematical physics. Complete formulations and proofs of all basic assertions are given, based on theorems of global analysis explained in the Appendix.