Vilasi G. Hamiltonian Dynamics
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Index
Algebra
Deformation
of
a
Lie algebra, 386
Grammann, 166
Lie algebra, 59, 219, 236
Lie algebra ideal, 214
Zform,
299
symplectic, 240
Ampere
Atlas, 129
Basis
anholonornic, 143
holonomic, 143,
181
tangent space,
134
vector space, 137
Arnphre, 296
Beltrami, 209
Bernoulli, 26
Betti, 208
Bianchi, 189
Bohr,
95
Coulomb, 104
Darboux, 240
Dirac, 100
Einstein, 190
Euier, 26
Faraday, 296
Galilei,
5
Biography
Gauss, 296
Hamilton,
14
Hertz,
18
Jacobi,
51
Lagrange, 401
Laplace,
83
Legendre, 40
Levi-Civita, 55
Lie, 66
Ljapunov, 356
Maxwell, 295
Newton, 6
Planck, 92
PoincarL-, 121
Poisson, 50
Riccati, 27
Ricci, 189
Riemann,
188
Schrodinger,
94
Stokes, 204
Bohr formula, 95
Brackets
commutator, 52
Lagrange, 76, 244
Lie, 156, 219
Nijenhuis, 192
Poisson, 50,
237,
244
base, 145
Bundle
437
438
Index
cotangent, 146, 239
fiber, 145
fiber bundle, 145
frame bundle, 147
Mobius, 147
principal fiber bundle, 147
structure group, 145
tangent, 143, 146
typical fiber, 145
Calculus of Variations, 24
Canonical
di~erentia~ 1-form, 239
symplectic structure, 239
Canonical system, 43
Chaotic behavior, 122
Constraints
anholonomic, 18
holonomic, 18
action-angle coordinates,
115,
250
action-angle type coordinates, 250
Lagrangian, 22
Coordinates
Covector, 137
Curve
brachjstochrone, 25
integral curve, 86, 148
Degrees of freedom, 22
Derivative
covariant, 185
exterior, 138, 172
exterior Nijenhuis, 387
Frechet, 29
Gateaux, 30, 339
Gradient,
31
Lie,
86,
151, 168
Li~Nijenhu~, 387
Distribution, 155
integral manifold, 156
involutive,
156
local basis, 155
maximal integral manifold, 156
Dual
Hodge, 200
General Hamiltonian, 57
Jacobi-Poisson,
59
ternary Jacobi-Poisson, 61
Dynamics
Equation
Einstein equations, 190
Euler equation, 292
Gel’fand-Levitan-Marchenko,
325
Hamilton-Jacobi equation, 96
Korteweg-de Vries,
311
Liouvi~~e, 345
modified Korteweg-de Vries, 317
nonlinear Schrijdinger, 380
Schrijdinger, 377
sine-Gordon, 343
Yang-Baxter, 388
Faraday
Flow
2-form,
299
Hamiltonian, 86
isospectral, 322
of
the vector field, 151
Foliation, 105, 157
leaf, 105,
157
Form
canonical 1-form, 410
closed, 173
differential k-form, 171
exact, 173
harmonic, 209
twisted, 197
Group
adjoint representation, 220
coadjoint representation, 225
Lie group, 211
local
Lie group, 213
group, 225
Hamiltonian
function, 233
Index
439
Hamiltonian vector field, 51
Hessian,
45
Identity
Bianchi,
189
Cartan,
177
contracted Bianchi, 189
Jacobi, 51
1ocalIy inertial frame,
9
absolute integral invariant, 246
complete integrd,
95
first integral, 237
integral invariant, 82
relative integral invariant, 247
H~ton-Jacobi method,
95
Inertial frame, 6,
8,
11
Integral
Integration
Involution, 54, 106
Jacobi
Jacobi identity, 51
Jacobi multip~er, 84
ternaxy Jacobi-Poisson bracket, 61
Lagrange
bracket, 76, 244
intrinsic equation,
241
Lagrangian,
13
function, 13
LaX
Lax
pair for KdV, 323
Lax Representation, 320
Lorentz force, 14
Manifold
level,
248
maximally symmetric, 184
orientable, 130, 195
submanifold, 154
sympiectic, 231
diff~morphism, 132
embedding, 154
Map
homeomorphism, 129
immersion, 154, 251
momentum, 273
PoincaxB, 122
differential equations, 297
geometrical equations, 300
phenomenological equations, 295
Maxwell
Operator
codifferential, 208
contraction, 174
endomorphism,
190
Laplace-Beltrami, 209
recursion operator,
259,
359
strong recursion operator, 368
weak
recursion
operator,
367
operators, 359
Poisson
Poisson bracket, 50
Poissonian action, 274
ternary Jacobi-Poisson bracket, 61
Einstein principle
of
equivalence, 9
Einstein principle
of
relativity, 9
Galilei principle
of
relativity, 5
Hamilton principle, 24, 34
Mac~-Einstein principle
of
relativity, 12
of
general covariance,
10
exterior, 166
interior, 176
tensor, 161
stereographic, 130
Principle
Product
projection
Scattering
inverse scattering method, 322
reflection coefficient, 322
transmission coefficient, 322
cotangent, 138
Space
440
Index
dual vector space, 136
phase space, 48, 241
reduced phase space, 277
state space, 47, 241
tangent, 134
vector space, 135
strongly structure, 342
structure, 231
weakly structure, 342, 343
Symplectic
Tensor, 161
field, 167
metric tensor, 166
metric tensor field,
181
Nijenhuis torsion, 192
Ricci, 189
Riemann, 188
scalar curvature, 189
super Nijenhuis torsion, 399
torsion, 187
Ado, 224
Cartan, 224
Daxboux, 240
Frobenius, 156
Hodge, 210
Jacobi-Poisson, 54
KAM,
121
Lee Hua-Chung, 68
Liouville, 107, 249
Liouville-Arnold, 362
rnonodromy, 224
Nother, 242, 245
Stokes, 204
virial, 46
Torus,
115,
116
Tkansformat ion
Theorem
~ea-prese~ing, 73
Biicklund,
315,
316
canonical, 66
completely canonical,
71,
244
generating function,
70
Hopf-Cole,
315
Legendre, 40, 242
push-forward, 140
symplectic, 73, 244
volume preserving, 75
pull-back,
140
Vector
complete vector field, 151
globally Hamiltonian vector field, 233
Killing vector field, 182
Laplace-Runge-Len2 vector,
90
left invariant vector field, 214
locally H~iltonian vector field, 233
Pauli,
93
tangent, 133, 134
vector field, 84, 140, 245