Lectures on Mechanics

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Cambridge University Press, 30 abr. 1992 - 254 páginas
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The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule.
 

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Índice

Introduction
1
12 Hamiltonian Formulation
3
13 Geometry Symmetry and Reduction
9
14 Stability
12
15 Geometric Phases
16
16 The Rotation Group and the Poincare Sphere
23
A Crash Course in Geometric Mechanics
27
22 The Flow of a Hamiltonian Vector Field
29
64 Cotangent Bundles General Case
120
65 Rigid Body Phases
122
66 Moving Systems
125
67 The Bead on the Rotating Hoop
127
Stabilization and Control
131
72 The Hamiltonian Structure with Feedback Controls
132
73 Feedback Stabilization of a Rigid Body with a Single Rotor
134
74 Phase Shifts
137

24 Lagrangian Mechanics
31
25 LiePoisson Structures
32
26 The Rigid Body
33
27 Momentum Maps
34
28 Reduction
36
29 Singularities and Symmetry
39
210 A Particle in a Magnetic Field
40
Cotangent Bundle Reduction
43
32 The Classical Water Molecule
46
33 The Mechanical Connection
50
34 The Geometry and Dynamics of Cotangent Bundle Reduction
54
35 Examples
59
36 Lagrangian Reduction
66
37 Coupling to a Lie group
72
Relative Equilibria
77
42 Cotangent Relative Equilibria
79
43 Examples
82
44 The Rigid Body
87
The EnergyMomentum Method
93
The Rigid Body
97
53 Block Diagonalization
101
54 The Normal Form for the Symplectic Structure
107
55 Stability of Relative Equilibria for the Double Spherical Pendulum
110
Geometric Phases
115
62 Reconstruction
117
63 Cotangent Bundle Phases a Special Case
119
75 The KaluzaKlein Description of Charged Particles
141
76 Optimal Control and YangMills Particles
144
Discrete reduction
147
81 Fixed Point Sets arid Discrete Reduction
149
82 Cotangent Bundles
155
83 Examples
157
84 SubBlock Diagonalization with Discrete Symmetry
162
85 Discrete Reduction of Dual Pairs
166
Mechanical Integrators
171
92 Limitations on Mechanical Integrators
175
93 Symplectic Integrators and Generating Functions
177
94 Symmetric Symplectic Algorithms Conserve J
178
95 EnergyMomentum Algorithms
180
96 The LiePoisson HamiltonJacobi Equation
182
The Free Rigid Body
186
98 Variational Considerations
187
Hamiltonian Bifurcation
189
102 The Role of Symmetry
196
103 The One to One Resonance and Dual Pairs
202
104 Bifurcations in the Double Spherical Pendulum
204
105 Continuous Symmetry Groups and Solution Space Singularities
205
106 The PoincareMelnikov Method
207
107 The Role of Dissipation
217
References
225
Index
250
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