Vilasi G. Hamiltonian Dynamics
Подождите немного. Документ загружается.
X
2.2.2
From Hamilton to Lagrange equations
2.2.3
Remarks
on
Hamilton's equations
2.3.1
The state space
2.3.2
The phase space
2.3.3
First integrals
2.3.4
The Poisson bracket
2.3.5
The Jacobi-Poisson theorem
2.4.1
General Hamiltonian dynamics
2.4.2
Jacobi-Poisson dynamics
2.4.3
More on the Poisson bracket
2.4.4
hrther generalizations of the Jacobi-Poisson dynamics
2.3
The Poisson Bracket and the Jacobi-Poisson Theorem
2.4
A More Compact Form
of
The Hamiltonian Dynamics
2.5
The Variational Principle for the Hamilton Equations
3
ans sf or mat ion
Theory
Contents
43
45
47
47
48
48
50
53
57
57
59
59
61
62
65
3.1
Canonical, Completely Canonical and Symplectic Transformations
3.1.1
Canonical transformations
3.1.2
A
general class of canonical transformations
3.1.3
Completely canonical transformations
3.1.4
Symplectic transformations
3.1.5
Area preserving tr~sforInations
3.1.6
Volume preserving transformation
3.2
A
New Characterization
of
Completely Canonical ~ansform~tio~
3.3
New Characterization
of
Sympletic Transformation
4.1
Integrals Invariants
of
a
Differential System
4.2
A
Primer
on
the Lie Derivative
4.3
The Kepler Dynamics
4
The Integration Methods
4.3.1
The Laplace-Runge-Lenz vector
4.3.2
The hydrogen atom
4.4
The Hamilton-Jacobi Integration Method
4.4.1
Remarks on the Hamilton-Jacobi equation
4.5
The
Ham~~ton-~aco~ Equation for the Kepler Potential
4.6
The Liouville Theorem on the Complete Integrability
4.6.1
Reduction
4.6.2
The Liouville theorem
4.6.3
Remarks on the Liouville theorem
65
65
69
71
73
73
75
75
80
81
81
85
89
90
92
95
99
101
105
105
107
114
Contents
xi
4.6.4 Action-angle coordinates 115
4.6.5 The action-angle coordinates for the Harmonic Oscillator 116
4.6.6 The Kepler dynamics in action-angle variables
118
4.6.7 The perturbations of integrable systems and the
KAM theorem 120
4.6.8 The Poinear6 representation 121
125
129
5.1 Differential Manifolds 129
5,2 Curves on a Differential Manifold
131
5.3 Tangent Space at
a
Point
133
5.3.1 Tangent vectors to a curve on
a
manifold
133
5.3.2 Tangent vectors to
a
manifold
134
5.4 A Digression on Vectors and Covectors
135
5.4.1 Vector space
135
5.4.2 Dual vector space 136
5.5 Cotangent Space
at
a
Point
138
5.6
Maps
Between Manifolds 139
5.7 Vector Fields 140
140
5.8
The
Tangent Bundle 143
5.9 General Definition of Fiber Bundle
144
5.9.1 More on the tangent bundle
145
5.9.2 Analysis of
two
bundles with
5‘’
as
base manifold 146
5.10
Integral Curves
of
a
Vector Field
148
5.11 The
Lie
Derivative 151
5.12 Submanifolds 154
5.12.1 The Frobenius theorem 155
6 Differentiai Forms 159
6.1 The Tensors 159
6.1.1 The pcovectors 164
6A.2 The exterior product 164
6.1.3 The metric tensor on
a
vector space
166
6.2 The Tensor Fields 167
6.2.1 The Lie derivative of
a
tensor field
167
6.2,2 The differential pforms
1
70
6.2.3
The exterior derivative 172
I1
Basic Ideas of Differential Geometry
5 M~folds and Tangent Spaces
5.7.1 Holonomic and anholonomic basis of vector fields
xii
Contents
6.2.4
Closed and exact differential forms
6.2.5
The contraction operator
ix
6.2.6
A different procedure
6.2.7
A
dual characterization of holonomic and
anholonomic basis
6.3
The Metric Tensor Field on a Manifold
6.3.1
Killing vector fields
6.3.2
Maximally symmetric manifolds
6.3.3
The Levi-Civita covariant derivative
6.3.4
The Riemann tensor field
6.3.5
The Ricci tensor and the scalar curvature
6.4.1
The Nijenhuis bracket of two mixed tensor fields
6.4
Endomorphisms Associated with a Mixed Tensor Field
7
Integration Theory
7.1
Orientable Manifolds
7.2
Integration on Orientable Manifolds
7.3
p-Vectors and Dual Tensors
7.4
Metric
o
Volume
=
Hodge Duality
7.5
Stokes Theorem
7.6
Gradient, Curl and Divergence
7.7
A Primer for Cohomology
7.8
Scalar Product of Differential p-Forms
7.8.1
Exterior codifferential
7.8.2
Hodge theorem
8
Lie Groups and Lie Algebras
8.1
Lie Groups
8.2
Building of a Lie Algebra F'rom
a
Lie Group
8.1.1
Local Lie groups
8.2.1
Lie algebras
8.2.2
Left invariant vector fields
8.2.3
The adjoint representation of a Lie group
8.2.4
The coadjoint representation of
a
Lie group
111
Geometry and Physics
9
Symplectic Manifolds and ~arn~~toni~ Systems
9.1
Symplectic Structures on
a
Manifold
9.2
Locally and Globally Hamiltonian Vector Fields
9.2.1
Integral curves of
a
Hamiltonian vector field
173
173
178
1
80
181
182
183
184
188
189
190
192
195
195
196
197
200
202
205
206
208
208
210
211
211
213
213
213
214
219
225
227
231
231
233
233
Contents
xiii
9.3
H~iltoni~ Flows
9.3.1
Lie algebras
of
~~iltonian vector fields and of
9.4
The Cotangent Bundle and Its Symplectic Structure
9.5
Revisited Analytical Mechanics
9.6
The Liouville Theorem
9.7
A New Characterization of Complete Integrability
Hamilton functions
9.6.1
The construction
of
the action-angles coordinates
9.7.1
From the Liouville integrability to invariant mixed
tensor fields
9.8
Applications
9.8.1
A
recursion operator for the rigid body dynamics
9.8.2
A
recursion operator for the Kepler dynamics
9.9.1
Compatible Poisson pairs
9.9
Pois~n-N~enhuis Structures
10
The Orbits Method
10.1
Wueed
Phase Space
10.2
Orbits of
a
Lie Group in the Coadjoint
presentation
10.3
The Rigid Body
10.3.1
The
space
and the
body
angular velocities
10.3.2
The
space
and the
body
angular momenta
10.4
Rigid Body Equations
11
Classical Electrodynamics
11.1
Maxwell’s Equations
11.2
Geometrical Identification of Fields on
@
11.3
Geometrical Ide~tification
of
Electroma~etic Field
in
Space-Time
11.3.1
The vector potential and the gauge transformation
11.3.2
Constitutive equations
11.3.3
The wave
qua ti on
11.3.4
Plane waves
IV
Integrable Field Theories
12
KdV
Equation
12.1
An Existence and U~~quen~ Theorem
12.2
Symmetries
12.2.1
Space-time symmetries
12.2.2
Biicklund tra~formation
235
236
239
241
248
250
252
259
261
261
264
267
267
271
271
280
288
289
291
291
295
295
297
299
300
301
303
304
307
311
311
314
314
315
xiv
Contents
12.3
Conservation Laws
12.3.1
Lax represent at ion
12.3.2
The inverse scattering method
12.4
KdV
as
a
Hamiltonian Dynamics
12.5
KdV
as
a
Completely Integrable Hamiltonian Dynamics
13.1
~otation and Generaiities
13.1.1
Backward to
KdV
13.2
Strongly and Weakly Symplectic Forms
13.3
Invariant Endomorphism
13
General Structures
13.3.1
Dynamical invariance
13.3.2
Nijenhuis torsion
13.3.3
Bidimensionality of eigenspaces of
T
(KdV
and
sG)
13.4
Invariant Endomorphisms and LiouvilIe’s Integrability
13.5
Recursion Operators in Dissipative Dynamics
13.5.1
The burgers hierarchy
14
Meaning and Existence of Recursion Operators
14.1
Integrable Systems
14.1.1
Alternative Hamiltonian descriptions for
14.1.2
Recursion operators
for
integrable systems
14.1.3
Liouvil~~Arnold integrable systems
15.1
A
Tensorial Version of the
Lax
Representation
integrable systems
15
Miscellanea
15.1.1
The
LR
of the harmonic oscillator
as
a parallel
15.1.2
The A-invariant tensor field for the harmonic oscillai
15.1.3
The A-invariant tensor field
for
KdV
15.1.4
The A-covariant tensor field
for
KdV
15.2.1
Comparison with the nonlinear Schrodinger equation
transport condition
15.2
Liouville Integrability of Schrodinger Equation
15.3
Integrable Systems
on
Lie
Croup
Coadjoint Orbits
15.4
Deformation of
a
Lie Algebra
15.4.1
Deformation
15.4.2
Lie-Nijenhuis and exterior-Nijenhuis derivatives
16
Integrability of Fermionic Dynamics
16.1
Recursion Operators in the Bosonic Case
318
320
322
328
330
337
337
340
342
343
346
347
347
347
352
353
359
360
361
363
365
369
369
372
;or
373
374
374
376
380
385
386
386
387
389
389
Contents
xv
16.2
Graded Differential Calculus
16.3
Poisson Supermanifold
A
Lagrange:
A
Short Biography
B Concerning the
Lie
Derivative
C
Concerning the Kepler Action Variables
D Concerning the Reduced Phase Space
E
On the Canonical D~fferenti~~
1-Form
F
Concerning Rigid
Body
Equations
G
The
Gelfand-Levitan-Marchenko
Equation
Bibliogr~phy
Index
16.3.1
Super ~~jenhuis torsion
390
394
399
401
404
406
409
410
413
414
427
437
Part
I
Analytical Mechanics
The aim of this part is
a
self-contained treatment of Classical Mechanics in
an advanced formulation. Many topics relevant to applications will not be
treated, since they can be found in excellent
te~tbooks.~~~~~~~~~~~~~~~~~~~~~~~~~
Our treatment is inspired by two important classical textbooks,
Lezioni
di
Meccanica Razionale
by
T.
Levi-Civita and
U.
Amaldi, and
The Analytical
Foundations
of
Celestial Mechanics
by
A.
Wintner.36*58
Definitions will be given for a
particle,
i.e. for
a
body whose space dimen-
sions
can
be neglected with respect to the dimensions
of
the
space
in which it
moves, and naturally extended to systems
of
particles and to continuous
sys-
tems
Cfields).
The simplicity
of
the formal extension from systems of particles
to fields, and the difficulties for
a
rigorous extension, will limit the treatment
to system
of
particles.
3