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Gardner R.B. — Method of Equivalence and Its Applications
Gardner R.B. — Method of Equivalence and Its Applications

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Название: Method of Equivalence and Its Applications

Автор: Gardner R.B.

Аннотация:

The ideas of Elie Cartan are combined with the tools of Felix Klein and Sophus Lie to present in this book the only detailed treatment of the method of equivalence. An algorithmic description of this method, which finds invariants of geometric objects under infinite dimensional pseudo-groups, is presented for the first time. As part of the algorithm, Gardner introduces several major new techniques. In particular, the use of Cartan's idea of principal components that appears in his theory of Repere Mobile, and the use of Lie algebras instead of Lie groups, effectively a linear procedure, provide a tremendous simplification. One must, however, know how to convert from one to the other, and the author provides the Rosetta stone to accomplish this. In complex problems, it is essential to be able to identify natural blocks in group actions and not just individual elements, and prior to this publication, there was no reference to block matrix techniques. The Method of Equivalence and Its Applications details ten diverse applications including Lagrangian field theory, control theory, ordinary differential equations, and Riemannian and conformal geometry. This volume contains a series of lectures, the purpose of which was to describe the equivalence algorithm and to show, in particular, how it is applied to several pedagogical examples and to a problem in control theory called state estimation of plants under feedback. The lectures, and hence the book, focus on problems in real geometry.


Язык: en

Рубрика: Математика/Геометрия и топология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1989

Количество страниц: 127

Добавлена в каталог: 16.04.2005

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Предметный указатель
$dSS^{-1}$      14 15 23 26 28 68
$G_2$      107
$H^0$      32
$h^{0, 2}$      32
$S_3$-Cartan lemma      27 28
$S_3$-lemma      27 69 98 100
$T_e$-valued Maurer — Cartan form      14
$\Gamma(m, p)$      83
$\pi$      31 32 39
$\tau_0$-modified coframe      38
Abstract computation      24
Adjoint representation      36
Automorphism of the e-structure      61 105
Betti number      73
Bilinear covariants      109 111 113 118 119
Cartan form      52
Cartan formula      43
Cartan — Kaehler Theorem      73 75 93
Cartan's Lemma      20 78 86 90 101
Cartan, Elie, equivalence problem of      1
Cartan, Elie, general equivalence theorem of      74
Characteristic variety      75
Christoffel form problem      69
Christoffel symbols      68
Coadjoint action      42 43
Coadjoint representation      36 105
Complete e-structure      72
Completely integrable system      16
Conformal geometry      2 12 76 83 97 102
Constant torsion      65
Contact geometry of a third-order ordinary differential equation      5
Contact transformation      7 8 76
Contragredient representation      35 41
Determinantal action      42
Diagonal subgroups $\Delta$      16 17
DIM      74 76 78 90
Discrete arithmetic subgroups      73
e-structure      57—59 61 65 69 71 72 93 101 104—107
e-structure, automorphism of      61
e-structure, complete      72
e-structure, equivalence of      59 66 68
e-structure, lifted      11 19
e-structure, order of      58 59 69
e-structure, rank of      58 59 69
Equivalence problem      53 77 83 84 88
Equivalence problem of e-structures      59
Equivalence problem of Elie Cartan      1
Equivalence problem, overdetermined      53 54
exp tX      43
Feedback diffeomorphisms      81
First fundamental theorem of Lie      14
First Lie algebra prolongation      31
First structure equations      34
First Whitehead theorem      73
First-order constant type      37 38
First-order Lagrangian      6
First-order normalization      39 102
Foliation      65
Frobenius theorem      16 60 67 73
Fundamental identity      119
Fundamental theorem of pseudo-Riemannian geometry      27
G(m)      83
G-space      11 22
General Equivalence Theorem of Cartan      74
Generalized geometry      89 94
Generalized orthogonal groups SO(p, q)      26
Geometry of control systems under feedback      8 81 83
Geometry of the integral      6 50
Geometry of time-fixed Newton's equations      4
Group-fiber representation      19 20
Groups up to right multiplications      59
H-reduction of type S      46
Hemihedrally isomorphic      116
Hilbert invariant integral      52
Identification of group of automorphisms on an e-structure      105
In involution      74 76 90 92 94 97 119
Independence conditions      75
Infinitesimal generators      19 67
Infinitesimal group actions      40 85
Integrable p-structure      71 72
Interior product      43
Intrinsic torsion      32
Invariant functions      24 53 116
Involutive      74
Levi-Civita connection      68
Levi-Civita connection form      29
Lie algebra valued compatible absorption      21 26 27 31 33 34
Lie derivative      43
Lie theorems, first fundamental      14 66
Lie theorems, second      19
Lie theorems, third fundamental      75
Lie theorems, third, local converse to      17
Lie's theorem only      78
Lifted equivalence problem      11 19
Local converse to the Third Lie Theorem      17
Loop A      47
M(n, R)-valued right invariant Maurer — Cartan form      15
Maurer — Cartan equations      16
Maurer — Cartan forms      14—17 19—21 24 26 28 43 73
Maurer — Cartan forms, $T_e$-valued      14
Maurer — Cartan forms, set of      14
Maximal Abelian subalgebras      105
Mod base      19 20 22
Modified forms      23
Natural absorptions      51 88
Natural “covariant derivatives”      57
Noncompact real form      107
One-dimensional time-fixed Newton's equations      47
One-parameter group of left translations      43
Operator L      22
Order of the e-structure      58
Overdetermined algorithm      8 54 81
Overdetermined equivalence problems      53 54
Overdetermined problem      3—5 81
Parametric calculation      25 52 89 102 103
Particle Lagrangians      7 52 83
Pfaffian equation      13 18 119
Poincare lemma      24
Principal components of order one      20 21 49 51 97
Principal left G-bundles      11
Pseudo-Riemannian metric      94
Quasilinear systems in involution      73
Rank of the e-structure      58
Reduced Cartan characters      74 76 77 90 98
Reduced tableau matrix      74 90 91
Reduction of the structure group      38 53 54 65
Regular of rank p      58
Repere Mobile      39
Restricted equivalence problem      57
Riemann-Christoffei curvature tensors      69
Riemann-Christoffel curvature matrix      29
Riemannian geometry      1 28 57 68
Right invariant vector field      43
Roots      106 107
Second Lie theorem      19
semisimple Lie algebras      105 106
Set of Maurer — Cartan forms      13
Similarity representation      41
SO(p, q) absorption theorem      27 29 97
Stabilizer type      43
Standard jet coordinates      4 6
State estimation of plants under feedback      81
Structure constants of G      14
Symbolic manipulation program      25 50
Tableau matrix      74
Technique of the graph      18 59 62 64 71 72
Tensor product of two representations      41
Third fundamental theorem of Lie      75
Third-order differential equation, contact geometry of      5 13
Third-order ordinary differential equations under contact transformations      5 102
Time-critical closed loop control      87
Time-fixed Newton's equations      4 55
Time-fixed Newton's equations, one-dimensional      47
Torsion coefficients      21 107
Torsion terms      21 23 77 104 105
Total differential equation      13
Transitive pseudogroup      75 76 78
Translational action      42
Underdetermincd problem      13 81
Web geometry      2 22 54 57 65
Zeroth-order invariant      53
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