Miron R., Hrimiuc D., Shimada H. — The Geometry of Hamilton and Lagrange Spaces :: Электронная библиотека попечительского совета мехмата МГУ

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Miron R., Hrimiuc D., Shimada H. — The Geometry of Hamilton and Lagrange Spaces

Miron R., Hrimiuc D., Shimada H. — The Geometry of Hamilton and Lagrange Spaces

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Название: The Geometry of Hamilton and Lagrange Spaces

Авторы: Miron R., Hrimiuc D., Shimada H.

Аннотация:

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.

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Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 338

Добавлена в каталог: 19.04.2005

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Предметный указатель

$\mathcal L$ -dual N-linear connections      168
$\mathcal L$ -dual nonlinear connections      163
$\mathcal L$ -duality      159
Absolute Energy      274
Adapted basis      15
Almost complex structure      59
Almost contact structure      320
Almost Hermitian structure      74
Almost Kahalerian model      156
Almost product strucutre      71
Almost symplectic structure      67
Autoparallel curves      19
Basis, adapted      15
Basis, local      5
Berwald spaces      50
Bianchi identities      26
Brackets, Lie brackets      242
Brackets, Poisson brackets      89
Bundle, differentiable      2
Bundle, dual      220
Bundle, k–osculator      219
Bundle, of higher order accelerations      219
Canonical dual semispray      231
Canonical k–semispray      391
Canonical metrical connections      130
Canonical nonlinear connections      10
Canonical N–linear connections      20
Canonical semispray      67
Canonical spray      40
Cartan metrical connection      146
Cartan nonlinear connection      144
Cartan spaces      139
Cartan special spaces      152
Christoffel symbols      39
Coefficients of a non–linear connection      127
Coefficients of an N–linear connection      129
Coefficients, dual of a non–linear connection      292
Connections, Berwald      50
Connections, Chern-Rund      49
Connections, Hashiguchi      49
Connections, Levi — Civita      59
Connections, nonlinear      10
Connections, N–metrical      75
Connections, remarkable      49
Coordinates of a point      2
Coordinates, transformation of      3
Covariant, differential      113
Covariant, h– and $v_\alpha-$       257
Covariant, h– and $w\alpha-$       257
Covariant, h– and $\nu$ – derivatives      22
Curvature of an N–linear connection      24
Deflection tensors      22
Direct decomposition      99
Distributions, horizontal      11
Distributions, vertical      3
Dual coefficients      292
Dual of       231
Dual semispray      297
d–vectors      28
d–vectors, d–tensors      107
d–vectors, Liouville d–vectors      298
Energy      66
Energy of superior order      294
Euler — Lagrange equations      294

f-related $\phi$ -connections      210
Finsler Cartan $\mathcal L$ -duality      173
Finsler metrics      32
Finsler spaces      32
Finsler special spaces      152
Fundamental function      32
Fundamental tensor of a Finsler space      32
Fundamental tensor of a higher order Lagrange spaces      290
General relativity      85
Generalized Lagrange spaces      84
Geodesics      39
Hamilton 1-form      130
Hamilton spaces      119
Hamilton vector field      130
Hamilton — Jacobi equations      125
Hamiltonian space of electrodynamics      134
Hamiltonian systems      125
Hamiltonian systems of higher order      288
Higher order Lagrange spaces      290
Higher order locally Minkowski spaces      182
Integral of action      65
Jacobi method      92
Jacobi — Ostrogradski momenta      294
Lagrange spaces      63
Lagrange — Hamilton duality      159
Lagrangian of higher order      290
Lagrangian, regular      64
Landsberg spaces      51
Legendre mapping      296
Lie derivative      5
Lift, homogeneous      57
Lift, horizontal      14
Lift, N–lift      123
Lift, Sasaki — Matsumoto      56
Liouville, 1–form      93
Liouville, vector fields      5
Locally Minkowski space      182
Lorentz equations      85
Main invariants      294
Nonlinear connection      127
N–linear connection      129
Poisson structure      92
Presymplectic structure      289
Projector, horizontal      14
Projector, vertical      14
Relativistic Optics      85
Ricci identities      25
Sections in $T^{*k}M$       227
Sections in $T^{k}M$       227
Spaces, Berwald      50
Spaces, Douglas      80
Spaces, Finsler      32
Spaces, Hamilton      124
Spaces, Kropina      181
Spaces, Landsberg      51
Spaces, Randers      177
Structure equations      29
Symplectic structures      89
Symplectic transformations      189
Symplectomorphism      208
Synge equations      293
Torsions      23 110
Variational problem      293
Whitney sum      11

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