拉格朗日幾何和哈密頓幾何：力學的應用(英文)

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

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