Audin M. — Spinning Tops: A Course on Integrable Systems :: Электронная библиотека попечительского совета мехмата МГУ

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Audin M. — Spinning Tops: A Course on Integrable Systems

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Название: Spinning Tops: A Course on Integrable Systems

Автор: Audin M.

Аннотация:

Since the time of Lagrange and Euler, it has been well known that an understanding of algebraic curves can illuminate the picture of rigid bodies provided by classical mechanics. A modern view of the role played by algebraic geometry has been established in recent years by many mathematicians. This book presents some of these modern techniques, which fall within the orbit of finite dimensional integrable systems. The main body of the text presents a rich assortment of methods and ideas from algebraic geometry prompted by classical mechanics, whilst in appendices the general, abstract theory is described. The methods are given a topological application, for the first time in book form, to the study of Liouville tori and their bifurcations. The book is based on courses for graduate students given by the author at Strasbourg University but the wealth of original ideas will make it also appeal to researchers.

Язык:

Рубрика: Физика/Динамические системы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 139

Добавлена в каталог: 22.07.2005

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Предметный указатель

$\Theta$ -divisor      10 85 113 117 126
$\vartheta$ -function      1 12 49 64 129
Abel — Jacobi mapping      11 38 85 87 88 116
Abelian function      1 6 10 112
Abelian variety      1 6 9 10 12 50 57 63 64 68 75 86 115 124
Action coordinates      5 13
Adjoint action      17 18 70 92 93 96 97
Affine structure      5 6 8 22 24
AKS theorem      4 8 12 20 33 61 63 79 98 100 101 102 104 108 111
Angle coordinates      5 8
Angular velocity      16
Anhannonic oscillator      5
Arnold — Lionville theorem      5 6 22
Azimuth      31
Bifurcations of Liouville tori      1 10 42 50 51 59 60 76
Birkhoff theorem      112
Calogero system      4
Calogero — Moser system      4
Calogero — Sutherland system      4
Casimir function      2 4 18 43 70 95 97 99
Centre-centre point      43
Clebsch rigid body      4
Coadjoint orbit      65 79 92 94—97 99—101
Coadjoint, action      4 92 94 94 94 100 102
Cookery recipe      63
Critical level      6 10 22 25 39 41 57 58 74 88
Curve, affine      34 72 104 114 115 128
Curve, complex      7 12 23 31 36—41 114 115
Curve, elliptic      12 22 28 29 34 38 46 63 67 71 73 75 113 116 122 123 125 127 129
Curve, hyperelliptie      1 48 49 54 72 73 80 83 115 127
Curve, real      12 38 88 113 118 119 121 123
Dimension-n rigid body      4
Divisor      13 34 35 37 38 54 56 67 73 80 83 104 108 110 115 116 116 117 121 123
Divisor, canonical      118
Divisor, effective      56 83 116
Divisor, general      83 86 118
Divisor, linear equivalence of      13 36 116 119 128
Eigenvector bundle      8 34 55 56 67 80 84 105 110 111 113
Eigenvector mapping      8—12 34 36 39 50 55 57 63 64 67 69 73 75 80 83 89 104 106 109 111 113 125
Elliptic function      1 6 11 65
Euler angles      31
Euler equations      4 10 16 17 65 66
Euler — Poinsot top      4 11 19 21 25
Euler — Poisson equations      17
Exotic SO(4)-top      4 62 71
Factorisation problem      see "Riemann problem"
First, integral      2 4 7 12 17 22 33 34 36 44 45 55 61 69—71 78
Focus-focus point      43
Free particle on an ellipsoid      4 9 30 73 124
Fundamental vector field      19 93 94—96
Gamier system      4
Gaudin system      4
Geodesic flow      4 9 30 73 124
Geodesies on an ellipsoid      4 9 30 73 124
Goldman functions      4
Goryachev — Chaplygin top      4 12 20 63 64
Grothendieck — Riemann — Roch theorem      see "Riemann — Roch theorem"
Hamilton equations      see "Hamiltonian system"
Hamiltonian Hopf bifurcation      43
Hamiltonian system      2 11 12 17 18 22 33 50 61 71 79 81 98 100—102 104 110 112 113
Hamiltonian vector field      2 10 19 22 29 57 73 101 113
harmonic oscillator      4
Heisenberg group      97
Henon — Heiles system      4 124
Holt potential      4
Inertia matrix      16 19 21 27 33 62
Integrable system      1—3 5 6 8—10 12 13 18 32 43 63 65 78 81 92 96 101 108 124

Intersection of quadrics      12 22 67 71
Involution theorem      54 70 98 99 100 112
Isospectral deformations      33 89 104 110
Jacobi identity      2 3 91—93 95 98
Jacobi matrix      78 89 102
Jacobian variety      10—12 36 37 46 48 50 55 64 69 86—89 110 113 114 115 116 118 119 122 123 127 129
Jacobian variety, generalised      34 36 58
Jeffrey — Weitsman system      4
Kepler problem      4
Kirchhoff rigid body      4
Kolosoff potential      5
Kowalevski top      1 5 9 11—13 19—21 45 50 58 59 62 63 76 129
Kowalevski — Painleve analysis      21 113
Lagrange momentum      19 29
Lagrange top      5 11 12 16 19 21 27 29 32 42 62 63
Lagrangian subspace      5
Laurent tail      108—110
Lax equation      6—11 13 32 33 37 44 45 50 52 60 62—66 77 78 82 103 104 108 110 113 118
Leibniz rule      2 3 91
Lie algebra      1 2 4 12 43 52 60 62 70 78 79 91—93 96—101 103 104 111 112
Linearisation of flows      5 6 8 9 22 24 36 48 50 64 104 110 111
Linearisation theorems      6 9 12 36 50 110
Liouville tori      1 5 6 10 25 27 29 30 42 50 51 56 58—60 76
Loop algebra      1 60
Momentum mapping      4 25 29 36 40—42 44 57 58 61 73 74
Moser system      5 44 73
Neumann problem      5 9
Normalisation      34 53 63 80 104 115 127 128
Nutation      31
Painleve analysis      see "Kowalevski — Painleve analysis"
Particle in a potential field      5
Pendulum      5
Period lattice      13 24 31 48 68 69 88 115 119 120 122—124
Pfaffian      70
Picard group      8 9 34 37 54 67 115 116 117 121
Poisson Lie group      92
Poisson structure      2 2—4 8 11 12 19 20 29 43 60 61 70 79 80 91 92 94—96 98—102 108
Poisson structure, Kirillov structure      12 19 61 70 91 94 95 98 100 101
Precession      31
Prym variety      9 10 12 54 56 63 74—76 124 125—127 129
R-matrix      60 61 63 98
Real part      9 10 24 28 31 37 38 40 41 49 50 56 58 59 66 69 71 72 75 86—89 113 118 119
Real structure      9 10 31 37 38 56 64 69 75 118 119 122
Regular level      5 6 9 12 22 25 29 30 36 37 40 41 43 50 56—58 63 73 88 113
Riemann problem      111 112
Riemann surface      114 115 118 129
Riemann — Hurwitz Theorem      114 115
Riemann — Roch theorem      11 34 55 75 85 105 106 114—116 117 118
Rigid body      1 4 9 11 12 15 16 21 22 29 62 65 66 69 125 129
Rigid body, Euler — Arnold      see "Rigid body free"
Rigid body, free      4 9 11 12 21 22 29 62 65 66 125
Ruijsenaars system      5
Sato Grassmannian      113
Skyscraper sheaf      108
Spectral curve      7 9 33 34 37 39 44 50 52 55 63 64 66 71 77 80 82 104 110 113 118
Spectral parameter      7 10 11 65 66 112 113
Spinning top      1 2 4 6 9—12 15 16 19 25 27—32 37 41 42 60 62 118
Spinning top, generalised      60
Steklov rigid body      5
Symplectic foliation      4 95
Symplectic leaf      4 20 29 65 69 80 92 94 96 97
Symplectic manifold      2—5 9 18 20 22 29 43 60 65 69 80 81 92 94 96 97 103 104
Symplectic reduction      29 61 62
Toda lattice      5 9 10 12 77 78 88 89 103 113
Two-body problem      5
Weierstrass $\wp$ -function      13 31 123 124
Yang — Baxter equation      98

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