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de Leon M., Rodrigues P.R. — Methods of differential geometry in analytical mechanics
de Leon M., Rodrigues P.R. — Methods of differential geometry in analytical mechanics



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Название: Methods of differential geometry in analytical mechanics

Авторы: de Leon M., Rodrigues P.R.

Аннотация:

The differential geometric formulation of analytical mechanics not only offers a new insight into Mechanics, but also provides a more rigorous formulation of its physical content from a mathematical viewpoint. Topics covered in this volume include differential forms, the differential geometry of tangent and cotangent bundles, almost tangent geometry, symplectic and pre-symplectic Lagrangian and Hamiltonian formalisms, tensors and connections on manifolds, and geometrical aspects of variational and constraint theories. The book may be considered as a self-contained text and only presupposes that readers are acquainted with linear and multilinear algebra as well as advanced calculus.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1989

Количество страниц: 495

Добавлена в каталог: 17.11.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(3, -1)$-structure      179
$(3, 1)$-structure      176
$(e)$-structures      102
$C^{\infty}$-atlas      6
$C^{\infty}$-compatible coordinate neighborhoods      6
$f$-basis      176
$f$-structure, integrable $f$-structure      178
$f$-structure, Riemannian metric adapted to an $f$-structure      178
$G$-connection      225
$G$-structures      100
$G$-structures, automorphism of $G$-structures      103
$G$-structures, integrable $G$-structures      105
$G$-structures, isomorphism of $G$-structures      103
$G$-structures, local automorphism of $G$-structures      104
$G$-structures, local isomorphism of $G$-structures      104
$G$-structures, locally isomorphic $G$-structures      103
$G$-structures, standard $G$-structures      104
$n$-submanifold property      12
$O(m)$-structures      102
$P$ — Euler — Lagrange vector field      414
$p$-form      43
$p$-form, closed $p$-form      71
$p$-form, exact $p$-form      71
$P$-second order differential equation      415
$P$-semispray      415
$r$-ary constraint manifold      401
$\phi$-basis      171
$\pi$-semibasic forms      331
$\pi_2$-semibasic forms      331
1-forms      28 29
Abelian Lie algebra      83
Action of a group, on the left      92
Action of a group, on the right      91
Adapted coframe, field      102
Adapted coframe, to a linear connection      131
Adapted frame      101
Adapted frame, field      101
Adapted frame, to a linear connection      131
Adapted frame, to an $f$-structure      176
Adapted frame, to an almost complex structure      152
Adapted frame, to an almost contact structure      171
Adapted frame, to an almost Hermitian structure      162
Adapted frame, to an almost product structure      148
Adapted frame, to an almost tangent structure      112
Adapted projector, to a presymplectic form on a vector space      233
Adjoint, operator $J*$      181
Adjoint, representation, of a Lie algebra      90
Adjoint, representation, of a Lie group      89
Admissible Lagrangian system      428
Affine space modeled on a vector space      138
Affine, bundle      138
Affine, morphism      138
Affine, space      138
Almost complex, connection      156
Almost complex, manifold      151
Almost complex, structure      151
Almost complex, structure, integrable almost complex structure      155
Almost complex, structure, on the cotangent bundle      249
Almost complex, structure, on the tangent bundle      165 166 218
Almost contact, compatible metric for an almost contact structure      170
Almost contact, metric structure      170
Almost contact, structure      169 172
Almost contact, structure, associated to a dynamical connection      342
Almost cosymplectic structure      172
Almost cotangent, manifolds      253
Almost cotangent, structures      253
Almost cotangent, structures, equivalent      259
Almost cotangent, structures, integrable      255
Almost cotangent, structures, regular      259
Almost product, connection      150
Almost product, integrable almost product structure      149
Almost product, manifold      147
Almost product, structure      147
Almost product, structure, on the cotangent bundle      248
Almost symplectic, connection      261
Almost symplectic, form      234
Almost symplectic, manifold      234
Almost symplectic, structure      234
Almost tangent, canonical almost tangent structure on $TN$      115
Almost tangent, connection      119
Almost tangent, integrable almost tangent structure      116
Almost tangent, manifold      111
Almost tangent, structure      111
Alternating mapping      38
Arc length of a curve      80
Automorphism preserving the symplectic form      229
Basic form on $T^*M$      368
Bianchi’s identities      79
Boundary, manifold with boundary      67
Boundary, manifold without boundary      67
Boundary, of $H^m$      66
Bracket product      82
Bracket product, of a derivation and a skew-derivation      45
Bracket product, of derivations      45
Bracket product, of skew-derivations      45
Bump function      10
Bundle      22
Bundle, base space of a bundle      22
Bundle, isomorphism      23
Bundle, morphism      23
Bundle, of linear frames      95
Bundle, projection of a bundle      22
Bundle, total space of a bundle      22
Bundle, trivial bundle      22
Campbell — Haussdorff formula      393 397
Canonical, 1-form      348
Canonical, coordinates      251
Canonical, equations      265 448
Canonical, transformation      236
Caplygin systems, generalized Caplygin systems      391
Cartan, formula      54
Cartan, symmetry      366
Cartan, Theorem      86
Cartesian product of vector bundles      25
Change of variables rule      65
Characteristic vector field      291
Christoffel components      76 201 223
Coisotropic, submanifold      236
Coisotropic, subspace      231
Coisotropic, vector subbundle      232
Compact support      10 44
Complete vector field      20
Complex, coordinates on a complex manifold      154
Complex, frame      152
Complex, manifold      154
Complex, structure      154
Components, of a contravariant tensor      34
Components, of a covariant tensor      34
Components, of a tensor field      36
Components, of a tensor of type $(r, s)$      34
Configuration, manifold      265 301
Configuration, space      443
Connection, form      221
Connection, in a fibred manifold      197
Connection, in a principal bundle      221
Connection, in a tangent bundle      200
Connection, path of a connection      204
Connection, symmetric connection      78
Conservative, connection      314
Conservative, force      441
Constants of motion      273
Constraint      295
Constraint, anholonomic constraint      297
Constraint, canonical constraint forces      296
Constraint, classical constraint      367
Constraint, holonomic constraint      297
Constraint, manifold      400
Constraint, system of constraints      295
Contact, form      169
Contact, manifold      169
Contact, metric structure      172
Contact, structure      169
Contraction operator      35
Contravariant, tensor of degree $r$      33
Contravariant, tensor space of degree $r$      33
Convex neighborhood      81
Coordinate, functions      5
Coordinate, neighborhood      5 67 154
Corank, of a 2-form      227 234
Corank, of a Poisson tensor field      285
Cotangent, bundle      23 30
Cotangent, vector space      28
Covariant derivative, of a curve      204
Covariant derivative, of a tensor field      78
Covariant derivative, of a vector field      76 203
Covariant tensor, of degree $s$      33
Covariant tensor, space of degree $s$      33
Critical section      351
Cross-section submanifold      349
Cup product      71
Curvature form      216 222
Curvature tensor      78 218
Curvature tensor, Riemannian curvature tensor      107
Curve      15
Darboux theorem      250
Darboux Theorem, Generalized Darboux Theorem      252
De Rham, algebra      71
de Rham, cohomology group      71
de Rham, theorem      72
Decomposition theorem      215
Derivation, of $\bar{C}^{\infty}(x)$      21
Derivation, of $\Lambda M$      45 46
Derivation, of $\Lambda M$, derivation of $\wedge M$ of degree $k$      45
Determinant, of a differentiable mapping      107
Determinant, of a linear map      107
Deviation vector field      430
Diffeomorphic manifold      8
Diffeomorphism      4 8
Diffeomorphism, preserving orientations      62
Diffeomorphism, preserving volume forms      62
Diffeomorphism, reversing orientations      62
Differentiable, function      3
Differentiable, invariant      72
Differentiable, manifold      6
Differentiable, mapping      4 8 66
Differentiable, structure      6
Differential equation, first order differential equation      18
Differential equation, second order differential equation      193 333
Differential, form      43
Differential, of a mapping      17
Distance function      81
Distinguished coordinates      60
distribution      55
Distribution, completely integrable distribution      55
Distribution, involutive distribution      55
Distribution, local basis of a distribution      55
Dynamical connection      336
Dynamical connection, components of a dynamical connection      336
Dynamical connection, path of a dynamical connection      340
Dynamical symmetry      366
Embedded submanifold      12
Embedding      12
Embedding, of principal bundles      97
Energy, conservation law      273 382 442
Energy, first form of the energy equation      442
Energy, function      263
Energy, second form of the energy equation      442
Energy, third form of the energy equation      443
Equivalence theorem      412
Euler — Lagrange, equations      304 335 445
Euler — Lagrange, equations, with constraints      368
Euler — Lagrange, vector field      304 336
Evolution space      330
Exponential map      84
Exterior, algebra      40
Exterior, derivative      47
Exterior, product      39
Extremal      351
Fiber derivative      329
Fibration, defined by an almost cotangent structure      258
Fibration, defined by an almost tangent structure      140
Fibre, of a bundle      22
Fibre, of a submersion      14
Fibred manifold      22
Final constraint manifold      401
First class, constrained manifold      275
First class, constraint function      406
First class, function of first class      406
First integral, of a system of differential equations      450
First integral, of a vector field      272
First-order, equivalent Hamiltonian systems      412
First-order, problem      400
Foliation      60
Foliation, leaves of a foliation      60
Force field      305
Frame bundle      95
Frobenius theorem      56
Fundamental form, of an almost contact metric structure      172
Fundamental form, of an almost Hermitian structure      163
Gauge equivalent Lagrangians      436
Gauge freedom      383
General linear group      7 82
General linear group, complex general linear group      85
Generalized, coordinates      443
Generalized, momenta      446
Generating function, of a canonical transformation      298
Generating function, of a Lagrangian submanifold      279
Geodesic, minimal geodesic      81
Geodesic, of a linear connection      76 204
Geodesically complete      77
Geometric algorithm of presymplectic systems      401
Germ of a function      21
Global dynamics      435
Grassman algebra      40
Green’s Theorem      70
Hamilton equations      265 293 448
Hamilton equations, intrinsical form of Hamilton equations      265
Hamilton equations, symplectic form of Hamilton equations      265
Hamilton — Jacobi method      279
Hamiltonian, energy      263 328
Hamiltonian, functional      350
Hamiltonian, regular Hamiltonian      369
Hamiltonian, system      263
Hamiltonian, system, with constraints      296
Hamiltonian, vector field      263 282
Hamiltonian, vector field, locally Hamiltonian vector field      263
Hamilton’s principle      351
Hamilton’s principle, modified Hamilton’s principle      354
Helmholtz conditions      323 395
Hermitian, almost Hermitian manifold      161
Hermitian, almost Hermitian structure      161
Hermitian, manifold      161
Hermitian, metric      161
Holonomic section      333
Homogeneous, connection      202
Homogeneous, form      189
Homogeneous, function      187
Homogeneous, vector field      188
Homogeneous, vector form      190
Hopf — Rinow theorem      81
Horizontal, bundle      197
Horizontal, component of a vector field      197
Horizontal, distribution      130 197
Horizontal, projector      197
Horizontal, subspace      197
Horizontal, tangent vector      197
Horizontal, vector field      197
Ideal, differential ideal      60
Ideal, of $\wedge M$      60
Image of a vector bundle homomorphism      25
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