拉格朗日几何和哈密顿几何——力学的应用
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作者(罗)拉杜·米龙
出版社哈尔滨工业大学出版社
ISBN9787560396576
出版时间2021-10
装帧平装
开本32开
定价48元
货号1202564218
上书时间2024-09-21
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目录
1 The Geometry of tangent manifold
1.1 The manifold TM
1.2 Semisprays on the manifold TM
1.3 Nonlinear connections
1.4 N-linear connections
1.5 Parallelism.Structure equations
2 Lagrange spaces
2.1 The notion of Lagrange space
2.2 Variational problem.Euler-Lagrange equations
2.3 Canonical semispray.Nonlinear connection
2.4 Hamilton-Jacobi equations
2.5 Metrical N-linear connections
2.6 The electromagnetic and gravitational fields
2.7 The almost Kahlerian model of a Lagrange space Ln
2.8 Generalized Lagrange spaces
3 Finsler Spaces
3.1 Finsler metrics
3.2 Geodesics
3.3 Cartan nonlinear connection
3.4 Cartan metrical connection
4 The Geometry of Cotangent Manifold
4.1 Cotangent bundle
4.2 Variational problem.Hamilton-Jacobi equations
4.3 Nonlinear connections
4.4 N-linear connections
4.5 Parallelism,paths and structure equations
5 Hamilton spaces
5.1 Notion of Hamilton space
5.2 Nonlinear connection of a Hamilton space
5.3 The canonical metrical connection of Hamilton space Hn
5.4 Generalized Hamilton Spaces GHn
5.5 The almost Kahlerian model of a Hamilton space
6 Cartan spaces
6.1 Notion of Caftan space
6.2 Canonical nonlinear connection of Ln
6.3 Canonical metrical connection of Ln
6.4 The duality between Lagrange and Hamilton spaces
7 The Geometry of the manifold TkM
7.1 The bundle of acceleration of order k≥1
7.2 The Liouville vector fields
7.3 Variational Problem
7.4 Semisprays.Nonlinear connections
7.5 The dual coefficients of a nonlinear connection
7.6 Prolongation to the manifold TkM of the Riemannian structures given on the base manifold M
7.7 N-linear connections on TkM
8 Lagrange Spaces of Higher-order
8.1 The spaces L(k)n=(M,L)
8.2 Examples of spaces L(k)n
8.3 Canonical metrical N-connection
8.4 The Riemannian (k-1)n-contact model of the space L(k)n
8.5 The generalized Lagrange spaces of order k
9 Higher-Order Finsler spaces
9.1 Notion of Finsler space of order k
10 The Geometry of k-cotangent bundle
10.1 Notion of k-cotangent bundle,T*kM
11 Riemannian mechanical systems
11.1 Riemannian mechanical systems
11.2 Examples of Riemannian mechanical systems
11.3 The evolution semispray of the mechanical system ΣR
11.4 The nonlinear connection of ΣR
11.5 The canonical metrical connection CΓ(N)
11.6 The electromagnetism in the theory of the Riemannian mechanical systems ΣR
11.7 The almost Hermitian model of the RMS ΣR
12 Finslerian Mechanical systems
12.1 Semidefinite Finsler spaces
12.2 The notion of Finslerian mechanical system
12.3 The evolution semispray of the system ΣF
12.4 The canonical nonlinear connection of the Finslerian mechanical systems ΣF
12.5 The dynamical derivative determined by the evolution nonlinear connection N
12.6 Metric N-linear connection of ΣF
12.7 The electromagnetism in the theory of the Finslerian mechanical systems ΣF
12.8 The almost Hermitian model on the tangent manifold TM of the Finslerian mechanical systems ΣF
13 Lagrangian Mechanical systems
13.1 Lagrange Spaces.Preliminaries
13.2 Lagrangian Mechanical systems,ΣL
13.3 The evolution semispray of ΣL
13.4 The evolution nonlinear connection of ΣL
13.5 Canonical N-metrical connection of ΣL.Structure equations
13.6 Electromagnetic field
13.7 The almost Hermitian model of the Lagrangian mechanical system ΣL
13.8 Generalized Lagrangian mechanical systems
14 Hamiltonian and Cartanian mechanical systems
14.1 Hamilton spaces.Preliminaries
14.2 The Hamiltonian mechanical systems
14.3 Canonical nonlinear connection of ΣH
14.4 The Cartan mechanical systems
15 Lagrangian,Finslerian and Hamiltonian mechanical systems of order k≥1
15.1 Lagrangian Mechanical systems of order k≥1
15.2 Lagrangian mechanical system of order k,ΣLk
……
编辑手记
内容摘要
本书是一部英文版的学术专著，中文书名可译为《拉格朗日几何和哈密顿几何：力学的应用》。本书的一个研究对象是拉杜·米龙首创的，如果说相近的，可能是Kahler流形。在当代数学的研究中，复流形的几何变得越来越重要了，特别是Kahler流形，所谓的Kahler流形是一个具有在典型复结构的作用下不变的黎曼度量的复流形，同时它的典型复结构在相应的黎曼联络下又是平行的。因此，Kahler流形是一类特殊的黎曼流形，具有更加丰富的几何结构，从而具有更加丰富多彩的几何性质。当然，Kahler流形可以从代数几何的角度进行研究，而且它是代数几何的主角，但是从微分几何的角度来了解它的几何结构和特征是十分重要的，也是研究Kahler流形的基础。
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