de Leon M., Rodrigues P.R. — Methods of differential geometry in analytical mechanics :: Электронная библиотека попечительского совета мехмата МГУ

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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:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1989

Количество страниц: 495

Добавлена в каталог: 17.11.2013

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Предметный указатель

Immersed submanifold      11
Immersion      11
Independent functions      275
Induced coordinates, in       27
Induced coordinates, in       30
Infinitesimal contact transformation      352
Inner automorphism of a Lie group      89
Integral, curve      18
Integral, lattice      115
Integral, manifold      55
Integral, of a form      66
Interior product      51 52
Inverse function theorem      4
Involution, canonical involution of       359 395
Involution, forms in involution      273
Involution, functions in involution      273
Isometry      104
Isomorphic, bundles      23
Isomorphic, vector bundles      23
Isotropic, submanifold      236
Isotropic, subspace      231
Isotropic, vector subbundle      232
Jacobi, field      129
Jacobi, identity      22 82
Jacobi, theorem      276
Jacobian matrix      4
Jet prolongation      332 352
Jets, -jets      451
Jets, of order       451
K $\ddot{a}$ hler, almost K $\ddot{a}$ hler manifold      165
K $\ddot{a}$ hler, form      163
K $\ddot{a}$ hler, manifold      165
Kernel, condition      435
Kernel, of a vector bundle homomorphism      25
Killing vector field      110
kinetic energy      329
Lagrange operator      378
Lagrangian, almost regular Lagrangian      411
Lagrangian, degenerate Lagrangian      302
Lagrangian, function      301 445
Lagrangian, function, admitting a map      362
Lagrangian, function, admitting a vector field      363
Lagrangian, homogeneous Lagrangian      306
Lagrangian, hyperregular Lagrangian      327 447
Lagrangian, irregular Lagrangian      302
Lagrangian, non-autonomous Lagrangian      330
Lagrangian, non-degenerate Lagrangian      302
Lagrangian, regular Lagrangian      302 447
Lagrangian, singular Lagrangian      302
Lagrangian, submanifold      236
Lagrangian, submanifold, generated by a Lagrangian      279 358
Lagrangian, subspace      231
Lagrangian, system      306
Lagrangian, system, with constraints      367
Lagrangian, time-dependent Lagrangian      330
Lagrangian, vector subbundle      232
Left, invariant form      83
Left, invariant vector field      83
Left, translation      83
Legendre transformation      327 449
Leibniz rule      21
Levi — Civita connection      79
Lie, algebra      82
Lie, algebra, automorphism      88
Lie, algebra, homomorphism      88
Lie, algebra, isomorphism      88
Lie, algebra, of a Lie group      84
Lie, bracket      22
Lie, derivative of a form      52
Lie, derivative of a tensor field      55
Lie, derivative of a vector field      106
Lie, group      81
Lie, group, acting effectively on the left      92
Lie, group, acting effectively, on the right      91
Lie, group, acting freely on the left      92
Lie, group, acting freely, on the right      91
Lie, group, automorphism      88
Lie, group, homomorphism      88
Lie, group, isomorphism      88
Lie, subalgebra      85
Lie, subgroup      85
Lie, symmetry      364
Lie, transformation group      91
Lift, complete lift of a distribution to       125
Lift, complete lift of a function to       121
Lift, complete lift of a linear connection to       126
Lift, complete lift of a tensor field of type to       124
Lift, complete lift of a tensor field of type to       244
Lift, complete lift of a tensor field of type to       245
Lift, complete lift of a tensor field of type to       124
Lift, complete lift of a vector field to       122
Lift, complete lift of a vector field to       243
Lift, horizontal lift of a distribution to       134
Lift, horizontal lift of a linear connection to       134
Lift, horizontal lift of a of a 1-form to       131
Lift, horizontal lift of a of a tensor field of type to       133
Lift, horizontal lift of a tensor field of type to       132
Lift, horizontal lift of a tensor field of type to       133
Lift, horizontal lift of a vector field to       130 201
Lift, natural lift      103
Lift, vertical lift of a a 1-form to       241
Lift, vertical lift of a function to       121
Lift, vertical lift of a function to       240
Lift, vertical lift of a tangent vector to       114
Lift, vertical lift of a tensor field of type to       121
Lift, vertical lift of a tensor field of type to       120
Lift, vertical lift of a vector field to       114
Lifts to the tangent bundle of order , of functions      453
Lifts to the tangent bundle of order , of tensor fields      453
Lifts to the tangent bundle of order , of vector fields      453
Linear connection      75 205 224
Linear connection, flat linear connection      79
Linear connection, in a tangent bundle      202
Linear connection, symmetric linear connection      78
Linear frame      94
Liouville, form      238
Liouville, property      287
Liouville, Theorem      271
Liouville, vector field      181 287
Liouville, vector field, locally Liouville      287
Local, coordinate system      5
Local, frame field      96
Locally finite covering      9
Maurer — Cartan equation      91
Maximal, $C^{\infty}$ -atlas      6
Maximal, integral manifold      60
Mechanical system      305
Mechanical system, conservative mechanical system      306 442
Mechanical system, non-conservative mechanical system      306
Mechanical system, with constraints      367
Metric in a vector bundle      106
Mixed tensor space of type       34
Momentum      327
Momentum conservation law      441
Newlander — Niremberg Theorem      156
Newton’s second law of motion      439
Nijenhuis tensor, of an almost complex structure      156
Nijenhuis tensor, of an almost tangent structure      116
Nijenhuis torsion      37
Noether symmetry      365
Noether’s theorem      363 390
Non-degenerate, 2-form      227 234
Non-degenerate, Poisson tensor field      285
Normal, almost contact metric structure      174
Normal, coordinates      391
Open submanifold      7
Orientable manifold      62
Orientation, induced orientation      68
Orientation, on vector spaces      61
Oriented vector space      61

Orthocomplement respect to asymplectic form      230
Orthogonal group      87 102
Orthogonal group, automorphism      109
Orthogonal group, complex orthogonal group      87
Orthogonal group, special orthogonal group      88
Orthonormal frame      103
Orthonormal frame, bundle      103
Paracompact topological space      9
Parallel, curve      204
Parallel, tensor field      78
Parallel, vector field      76
Parallelizable manifold      102 1
Parameter group      19 1-
Parameter group of local transformations      19 1
Parameter group of transformations      19 1-
Parameter group of transformations, infinitesimal generator of a      1-
Parameter group of transformations, local      1-
Parameter group of transformations, time-dependent      249 1-
Parameter subgroup of a Lie group      84
Partial derivative      3
Partition of unity subordinate to an atlas      10
Phase space, of momenta      265
Phase space, of velocities      301
Poincar $\acute{e}$ lemma      72
Poincar $\acute{e}$ — Cartan, 1-form      334
Poincar $\acute{e}$ — Cartan, 2-form      334
Poincar $\acute{e}$ — Cartan, form of higher order      456
Poisson, bracket of 1-forms      268
Poisson, bracket of functions      267 282 450
Poisson, manifold      282 284
Poisson, map      298
Poisson, structure      282
Poisson, tensor field      283
Polar decomposition      109
Positive definite automorphism      109
Positively oriented, basis      62 68
Positively oriented, coordinate neighborhood      63
potential energy      441
Potential of a semibasic vector form      195
Presymplectic, forms on manifolds      251
Presymplectic, forms on vector spaces      233
Presymplectic, manifold      251
Presymplectic, structure      251
Presymplectic, vector space      233
Principal bundle      93
Principal bundle, base space of a principal bundle      94
Principal bundle, bundle space of a principal bundle      94
Principal bundle, homomorphism      96
Principal bundle, projection of a principal bundle      93
Principal bundle, section of a principal bundle      96
Principal bundle, total space of a principal bundle      94
Principal bundle, transition functions of a principal bundle      97
Principal bundle, trivial principal bundle      94
Principal bundle, vertical subspace of a principal bundle      94
Principal bundle, vertical tangent vector to a principal bundle      94
Principal subbundle      97
Product manifold      7
Projectable vector field      351
Prolongable vector field      430
Prolongation      1-
Prolongation, -jet prolongation      452
Prolongation, canonical prolongation      333 453
Quotient manifold      14
Range condition      399
Rank, 2-form of maximal rank      227 234
Rank, of a 2-form      227 234
Rank, of a function      4
Rank, of a mapping      8
Rank, of a Poisson tensor field      285
Rank, theorem      5
Rank, theorem for manifolds      9
Real projective space      14
Reduced bundle      97
Reduced Lagrangian system      436
Reeb vector field      169 290
refinement      9
Regular, 2-form      227
Regular, condition for Poisson brackets      274
Regular, consistent constraints      401
Regular, Lagrangian dynamical system      322
Regular, Lagrangian vector field      379
Regular, non-conservative regular Lagrangi vector field      388
Representation, of a Lie algebra      88
Representation, of a Lie group      88
Ricci tensor      108
Riemannian, connection      79
Riemannian, induced Riemannian metric      107
Riemannian, metric      37
Riemannian, prolongation      219
Riemannian, pseudo — Riemannian metric      38
Riemannian, structure      102
Right translation      83
Sasaki metric      135
Sasakian almost contact metric structure      175
Scalar curvature      108
Schouten bracket      283
Second class, function of second class      406
Second-order problem      410
Section of a bundle      22
Sectional curvature      108
Semi-prolongable vector fields      431
Semi-prolongable vector fields, -equivalent semi-prolongable vector fields      432
Semibasic forms      190
Semibasic forms, algebra of semibasic forms      191
Semibasic forms, on       368
Semibasic forms, potential of semibasic forms      195
Semibasic vector form      193
Semispray      193 333
Semispray, associated semispray      206 337
Semispray, deviation of a semispray      194
Semispray, path of a semispray      194
Semispray, solution of a semispray      194
Shuffles permutations      39
Signature, of a pseudo — Riemannian metric      145
Signature,of a symmetric tensor of type (0, 2)      145
Simple connection      314
Skew — Hermitian matrix      87
Skew-derivation of $\wedge M$       45
Skew-derivation of degree       45
Skew-symmetric matrix      87
Source projection      452
Special, linear group      87
Special, real special linear group      87
Special, symplectic manifold      357
Special, unitary group      86
Spray      195
Spray, quadratic spray      195
Standard orientation of       62
Stoke’s Theorem      68
Strong horizontal, bundle      338
Strong horizontal, lift      338
Strong horizontal, projector      339
Strong horizontal, subspace      338
Strong horizontal, vector field      338
Strong, -torsion      341
Strong, $\bar{J}$ -torsion      341
Strong, curvature      341
Structure      176
Structure, 1-form      348
Structure, constants      91
Structure, equation      222
Structure, group of a principal bundle      94
Structure, group of a principal bundle, reducible group of a principal bundle      97
Structure, group of a principal bundle, reduction of the group of a principal bundle      97
Structure, metric -structure      178
Structure, partially integrable -structure      178
Submersion      13
Support, of a form      44
Support, of a function      10
Symmetric, automorphism      109

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